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Server: Apache/2.4.46 (Win64) OpenSSL/1.1.1j PHP/8.4.25
System: Windows NT DESKTOP-4TAV2RJ 10.0 build 19045 (Windows 10) AMD64
User: fred (0)
PHP: 8.4.25
Disabled: NONE
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File: C:/Users/fred/anaconda3/Lib/site-packages/mkl_random/mklrand.cp312-win_amd64.pyd
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    RandomState(seed=None, brng='MT19937')

    Container for the Intel(R) MKL-powered (pseudo-)random number generators.

    `RandomState` exposes a number of methods for generating random numbers
    drawn from a variety of probability distributions. In addition to the
    distribution-specific arguments, each method takes a keyword argument
    `size` that defaults to ``None``. If `size` is ``None``, then a single
    value is generated and returned. If `size` is an integer, then a 1-D
    array filled with generated values is returned. If `size` is a tuple,
    then an array with that shape is filled and returned.

    *Compatibility Notice*
    This version of numpy.random has been rewritten to use MKL's vector
    statistics functionality, that provides efficient implementation of
    the MT19937 and many other basic psuedo-random number generation
    algorithms as well as efficient sampling from other common statistical
    distributions. As a consequence this version is NOT seed-compatible with
    the original numpy.random.

    Parameters
    ----------
    seed : {None, int, array_like}, optional
        Random seed initializing the pseudo-random number generator.
        Can be an integer, an array (or other sequence) of integers of
        any length, or ``None`` (the default).
        If `seed` is ``None``, then `RandomState` will try to read data from
        ``/dev/urandom`` (or the Windows analogue) if available or seed from
        the clock otherwise.
    brng : {'MT19937', 'SFMT19937', 'MT2203', 'R250', 'WH', 'MCG31', 'MCG59',
            'MRG32K3A', 'PHILOX4X32X10', 'NONDETERM', 'ARS5'}, optional
        basic pseudo-random number generation algorithms, or non-deterministic
        hardware-based generator, provided by Intel MKL. The default choice is 
        'MT19937' - the Mersenne Twister generator.

    Notes
    -----
    The Python stdlib module "random" also contains a Mersenne Twister
    pseudo-random number generator with a number of methods that are similar
    to the ones available in `RandomState`. `RandomState`, besides being
    NumPy-aware, has the advantage that it provides a much larger number
    of probability distributions to choose from.

    References
    -----
    MKL Documentation: https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl.html

    2147483648429496729542949672960x80000000000000000x10000000000000000-9223372036854775808builtinstypenumpydtypeflatiterbroadcastndarraynumberintegersignedintegerunsignedintegerinexactfloatingcomplexfloatingflexiblecharacterufuncmklrandInterpreter change detected - this module can only be loaded into one interpreter per process.name__loader__loader__file__origin__package__parent__path__submodule_search_locationsModule 'mklrand' has already been imported. Re-initialisation is not supported.cython_runtime__builtins__mkl_random.mklrandinit mkl_random.mklrandname '%U' is not defined while calling a Python objectNULL result without error in PyObject_Callcalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseException'%.200s' object is unsliceableat leastat mostexactlys%.200s() takes %.8s %zd positional argument%.1s (%zd given)%s() got multiple values for keyword argument '%U'%.200s() keywords must be strings%s() got an unexpected keyword argument '%U'cannot fit '%.200s' into an index-sized integer'%.200s' object is not subscriptabletoo many values to unpack (expected %zd)need more than %zd value%.1s to unpackMissing type objectCannot convert %.200s to %.200sstrings are too large to concatlocal variable '%s' referenced before assignmentassignmentdeletion'%.200s' object does not support slice %.10scannot import name %Sbase class '%.200s' is not a heap typeextension type '%.200s' has no __dict__ slot, but base type '%.200s' has: either add 'cdef dict __dict__' to the extension type or add '__slots__ = [...]' to the base typeinvalid vtable found for imported typemultiple bases have vtable conflict: '%.200s' and '%.200s'%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%s.%s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject_cython_3_0_10Shared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompilingkeywords must be strings__name__ must be set to a string object__qualname__ must be set to a string objectfunction's dictionary may not be deletedsetting function's dictionary to a non-dict__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function calls__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function calls__annotations__ must be set to a dict objectfunc_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutine__module__Bad call flags for CyFunction<cyfunction %U at %p>%.200s() takes no arguments (%zd given)%.200s() takes exactly one argument (%zd given)%.200s() takes no keyword argumentsunbound method %.200S() needs an argument%.200s() needs an argument_cython_3_0_10.cython_function_or_method%s (%s:%d)value too large to convert to intvalue too large to convert to npy_boolcan't convert negative value to npy_boolvalue too large to convert to npy_int8value too large to convert to npy_int16value too large to convert to npy_int32value too large to convert to npy_uint8can't convert negative value to npy_uint8value too large to convert to npy_uint16can't convert negative value to npy_uint16can't convert negative value to npy_uint32can't convert negative value to npy_uint64value too large to convert to irk_brng_tcan't convert negative value to unsigned intvalue too large to convert to longcan't convert negative value to size_tdoes not matchcompile time Python version %d.%d of module '%.100s' %s runtime version %d.%d__int__ returned non-int (type %.200s).  The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__%.4s__ returned non-%.4s (type %.200s)intan integer is required_rand_int32unsafefhighRandomState.chisquareRandomState.randintWHfloatingmlistobUnsupported dtype "%s" for randintjdfnum <= 0cdfRandomState.logseriesstandard_normalngood < 0RandomState.weibull (line 3607)RandomState.hypergeometric (line 5004)
        _rand_int32(self, low, high, size)

        Return random np.int32 integers between `low` and `high`, inclusive.

        Return random integers from the "discrete uniform" distribution in the
        closed interval [`low`, `high`]. On entry the arguments are presumed
        to have been validated for size and order for the np.int32 type.

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution.
        high : int
            Highest (signed) integer to be drawn from the distribution.
        size : int or tuple of ints
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : python scalar or ndarray of np.int32
              `size`-shaped array of random integers from the appropriate
              distribution, or a single such random int if `size` not provided.

        multivariate_normal
        set_state(state)

        Set the internal state of the generator from a tuple.

        For use if one has reason to manually (re-)set the internal state of the
        chosen pseudo-random number generating algorithm.

        Parameters
        ----------
        state : tuple(str, bytes)
            The `state` tuple has the following items:

            1. the string, defaulting to 'MT19937', specifying the
               basic psedo-random number generation algorithm.
            2. a bytes object holding content of Intel MKL's stream for the
               given BRNG.

        Returns
        -------
        out : None
            Returns 'None' on success.

        See Also
        --------
        get_state

        Notes
        -----
        `set_state` and `get_state` are not needed to work with any of the
        random distributions in NumPy. If the internal state is manually altered,
        the user should know exactly what he/she is doing.

        For backwards compatibility, the form (str, array of 624 uints, int) is
        also accepted although in such a case keys are used to seed the generator,
        and position index pos is ignored: ``state = ('MT19937', keys, pos)``.

        References
        ----------
        MKL Documentation: https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl.html

        
        tomaxint(size=None)

        Return a sample of uniformly distributed random integers in the interval
        [0, ``np.iinfo("long").max``].

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Drawn samples, with shape `size`.

        See Also
        --------
        randint : Uniform sampling over a given half-open interval of integers.
        random_integers : Uniform sampling over a given closed interval of
            integers.

        Examples
        --------
        >>> RS = np.random_random_intel.RandomState() # need a RandomState object
        >>> RS.tomaxint((2,2,2))
        array([[[1170048599, 1600360186],
                [ 739731006, 1947757578]],
               [[1871712945,  752307660],
                [1601631370, 1479324245]]])
        >>> import sys
        >>> sys.maxint
        2147483647
        >>> RS.tomaxint((2,2,2)) < sys.maxint
        array([[[ True,  True],
                [ True,  True]],
               [[ True,  True],
                [ True,  True]]], dtype=bool)

        _rand_uint8This function is deprecated. Please call randint({low}, {high} + 1) instead
        _rand_uint32(self, low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        uRandomState._rand_uint64lengthfmustream_bufretpermutationlow_rand_bool
        noncentral_f(dfnum, dfden, nonc, size=None)

        Draw samples from the noncentral F distribution.

        Samples are drawn from an F distribution with specified parameters,
        `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
        freedom in denominator), where both parameters > 1.
        `nonc` is the non-centrality parameter.

        Parameters
        ----------
        dfnum : int
            Parameter, should be > 1.
        dfden : int
            Parameter, should be > 1.
        nonc : float
            Parameter, should be >= 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : scalar or ndarray
            Drawn samples.

        Notes
        -----
        When calculating the power of an experiment (power = probability of
        rejecting the null hypothesis when a specific alternative is true) the
        non-central F statistic becomes important.  When the null hypothesis is
        true, the F statistic follows a central F distribution. When the null
        hypothesis is not true, then it follows a non-central F statistic.

        References
        ----------
        .. [1] Weisstein, Eric W. "Noncentral F-Distribution."
               From MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/NoncentralF-Distribution.html
        .. [2] Wikipedia, "Noncentral F distribution",
               http://en.wikipedia.org/wiki/Noncentral_F-distribution

        Examples
        --------
        In a study, testing for a specific alternative to the null hypothesis
        requires use of the Noncentral F distribution. We need to calculate the
        area in the tail of the distribution that exceeds the value of the F
        distribution for the null hypothesis.  We'll plot the two probability
        distributions for comparison.

        >>> dfnum = 3 # between group deg of freedom
        >>> dfden = 20 # within groups degrees of freedom
        >>> nonc = 3.0
        >>> nc_vals = mkl_random.noncentral_f(dfnum, dfden, nonc, 1000000)
        >>> NF = np.histogram(nc_vals, bins=50, normed=True)
        >>> c_vals = mkl_random.f(dfnum, dfden, 1000000)
        >>> F = np.histogram(c_vals, bins=50, normed=True)
        >>> plt.plot(F[1][1:], F[0])
        >>> plt.plot(NF[1][1:], NF[0])
        >>> plt.show()

        RuntimeWarningRandomState.multinormal_cholesky (line 5380)Box-Mulleraddnonc < 0lnbaduniquel__RandomState_ctorkPHILOX4X32X10BoxMullera <= 0odflogistic
        leapfrog(k, nstreams)

        Initializes the current state generator using leap-frog method,
        if supported for the basic random pseudo-random number generation
        algorithm.
        _method_alias_dict_gaussianhiRandomState.rayleigh (line 4283)enableaRandomState.exponentialarray_int_datacline_in_traceback
        multivariate_normal(mean, ch, size=None, method='ICDF')

        Draw random samples from a multivariate normal distribution.

        The multivariate normal, multinormal or Gaussian distribution is a
        generalization of the one-dimensional normal distribution to higher
        dimensions.  Such a distribution is specified by its mean and
        covariance matrix, specified by its lower triangular Cholesky factor.
        These parameters are analogous to the mean
        (average or "center") and standard deviation, or "width,"
        of the one-dimensional normal distribution.

        Parameters
        ----------
        mean : 1-D array_like, of length N
            Mean of the N-dimensional distribution.
        ch : 2-D array_like, of shape (N, N)
            Cholesky factor of the covariance matrix of the distribution. Only lower-triangular
            part of the matrix is actually used.
        size : int or tuple of ints, optional
            Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
            generated, and packed in an `m`-by-`n`-by-`k` arrangement.  Because
            each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
            If no shape is specified, a single (`N`-D) sample is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        Notes
        -----
        The mean is a coordinate in N-dimensional space, which represents the
        location where samples are most likely to be generated.  This is
        analogous to the peak of the bell curve for the one-dimensional or
        univariate normal distribution.

        Covariance indicates the level to which two variables vary together.
        From the multivariate normal distribution, we draw N-dimensional
        samples, :math:`X = [x_1, x_2, ... x_N]`.  The covariance matrix
        element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
        The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
        "spread").

        Instead of specifying the full covariance matrix, popular
        approximations include:

          - Spherical covariance (*cov* is a multiple of the identity matrix)
          - Diagonal covariance (*cov* has non-negative elements, and only on
            the diagonal)

        This geometrical property can be seen in two dimensions by plotting
        generated data-points:

        >>> mean = [0, 0]
        >>> cov = [[1, 0], [0, 100]]  # diagonal covariance

        Diagonal covariance means that points are oriented along x or y-axis:

        >>> import matplotlib.pyplot as plt
        >>> x, y = mkl_random.multivariate_normal(mean, cov, 5000).T
        >>> plt.plot(x, y, 'x')
        >>> plt.axis('equal')
        >>> plt.show()

        Note that the covariance matrix must be positive semidefinite (a.k.a.
        nonnegative-definite). Otherwise, the behavior of this method is
        undefined and backwards compatibility is not guaranteed.

        References
        ----------
        .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
               Processes," 3rd ed., New York: McGraw-Hill, 1991.
        .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
               Classification," 2nd ed., New York: Wiley, 2001.

        Examples
        --------
        >>> mean = (1, 2)
        >>> cov = [[1, 0], [0, 1]]
        >>> x = mkl_random.multivariate_normal(mean, cov, (3, 3))
        >>> x.shape
        (3, 3, 2)

        The following is probably true, given that 0.6 is roughly twice the
        standard deviation:

        >>> list((x[0,0,:] - mean) < 0.6)
        [True, True]

        resarroffmkl_random\mklrand.pyxReturn a RandomState instance.
    This function exists solely to assist (un)pickling.
    Note that the state of the RandomState returned here is irrelevant, as this function's
    entire purpose is to return a newly allocated RandomState whose state pickle can set.
    Consequently the RandomState returned by this function is a freshly allocated copy
    with a seed=0.
    See https://github.com/numpy/numpy/issues/4763 for a detailed discussion
    left > modefdfthreadingchoiceomulnsamplerightn_uniqopstandard_cauchyRandomState.randravelactualepsnoncentral_chisquareRandomState.pareto (line 3503)sum(pvals[:-1]) > 1.0ototlongRandomState.set_statechoose_methodstandard_exponentialRandomState.randn (line 2213)LockuniformRandomState.uniform (line 2071)RandomState.dirichlet (line 5648)
        choice(a, size=None, replace=True, p=None)

        Generates a random sample from a given 1-D array

                .. versionadded:: 1.7.0

        Parameters
        -----------
        a : 1-D array-like or int
            If an ndarray, a random sample is generated from its elements.
            If an int, the random sample is generated as if a was np.arange(n)
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        replace : boolean, optional
            Whether the sample is with or without replacement
        p : 1-D array-like, optional
            The probabilities associated with each entry in a.
            If not given the sample assumes a uniform distribution over all
            entries in a.

        Returns
        --------
        samples : 1-D ndarray, shape (size,)
            The generated random samples

        Raises
        -------
        ValueError
            If a is an int and less than zero, if a or p are not 1-dimensional,
            if a is an array-like of size 0, if p is not a vector of
            probabilities, if a and p have different lengths, or if
            replace=False and the sample size is greater than the population
            size

        See Also
        ---------
        randint, shuffle, permutation

        Examples
        ---------
        Generate a uniform random sample from np.arange(5) of size 3:

        >>> mkl_random.choice(5, 3)
        array([0, 3, 4])
        >>> #This is equivalent to mkl_random.randint(0,5,3)

        Generate a non-uniform random sample from np.arange(5) of size 3:

        >>> mkl_random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
        array([3, 3, 0])

        Generate a uniform random sample from np.arange(5) of size 3 without
        replacement:

        >>> mkl_random.choice(5, 3, replace=False)
        array([3,1,0])
        >>> #This is equivalent to mkl_random.permutation(np.arange(5))[:3]

        Generate a non-uniform random sample from np.arange(5) of size
        3 without replacement:

        >>> mkl_random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
        array([2, 3, 0])

        Any of the above can be repeated with an arbitrary array-like
        instead of just integers. For instance:

        >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
        >>> mkl_random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
        array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'],
              dtype='|S11')

        
        standard_gamma(shape, size=None)

        Draw samples from a standard Gamma distribution.

        Samples are drawn from a Gamma distribution with specified parameters,
        shape (sometimes designated "k") and scale=1.

        Parameters
        ----------
        shape : float
            Parameter, should be > 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The drawn samples.

        See Also
        --------
        scipy.stats.distributions.gamma : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Gamma distribution is

        .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},

        where :math:`k` is the shape and :math:`\theta` the scale,
        and :math:`\Gamma` is the Gamma function.

        The Gamma distribution is often used to model the times to failure of
        electronic components, and arises naturally in processes for which the
        waiting times between Poisson distributed events are relevant.

        References
        ----------
        .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
               Wolfram Web Resource.
               http://mathworld.wolfram.com/GammaDistribution.html
        .. [2] Wikipedia, "Gamma-distribution",
               http://en.wikipedia.org/wiki/Gamma-distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> shape, scale = 2., 1. # mean and width
        >>> s = mkl_random.standard_gamma(shape, 1000000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> import scipy.special as sps
        >>> count, bins, ignored = plt.hist(s, 50, normed=True)
        >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ \
        ...                       (sps.gamma(shape) * scale**shape))
        >>> plt.plot(bins, y, linewidth=2, color='r')
        >>> plt.show()

        
        lognormal(mean=0.0, sigma=1.0, size=None, method='ICDF')

        Draw samples from a log-normal distribution.

        Draw samples from a log-normal distribution with specified mean,
        standard deviation, and array shape.  Note that the mean and standard
        deviation are not the values for the distribution itself, but of the
        underlying normal distribution it is derived from.

        Parameters
        ----------
        mean : float
            Mean value of the underlying normal distribution
        sigma : float, > 0.
            Standard deviation of the underlying normal distribution
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER

        Returns
        -------
        samples : ndarray or float
            The desired samples. An array of the same shape as `size` if given,
            if `size` is None a float is returned.

        See Also
        --------
        scipy.stats.lognorm : probability density function, distribution,
            cumulative density function, etc.

        Notes
        -----
        A variable `x` has a log-normal distribution if `log(x)` is normally
        distributed.  The probability density function for the log-normal
        distribution is:

        .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
                         e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}

        where :math:`\mu` is the mean and :math:`\sigma` is the standard
        deviation of the normally distributed logarithm of the variable.
        A log-normal distribution results if a random variable is the *product*
        of a large number of independent, identically-distributed variables in
        the same way that a normal distribution results if the variable is the
        *sum* of a large number of independent, identically-distributed
        variables.

        References
        ----------
        .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal
               Distributions across the Sciences: Keys and Clues,"
               BioScience, Vol. 51, No. 5, May, 2001.
               http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
        .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme
               Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, sigma = 3., 1. # mean and standard deviation
        >>> s = mkl_random.lognormal(mu, sigma, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 100, normed=True, align='mid')

        >>> x = np.linspace(min(bins), max(bins), 10000)
        >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
        ...        / (x * sigma * np.sqrt(2 * np.pi)))

        >>> plt.plot(x, pdf, linewidth=2, color='r')
        >>> plt.axis('tight')
        >>> plt.show()

        Demonstrate that taking the products of random samples from a uniform
        distribution can be fit well by a log-normal probability density
        function.

        >>> # Generate a thousand samples: each is the product of 100 random
        >>> # values, drawn from a normal distribution.
        >>> b = []
        >>> for i in range(1000):
        ...    a = 10. + mkl_random.random(100)
        ...    b.append(np.product(a))

        >>> b = np.array(b) / np.min(b) # scale values to be positive
        >>> count, bins, ignored = plt.hist(b, 100, normed=True, align='mid')
        >>> sigma = np.std(np.log(b))
        >>> mu = np.mean(np.log(b))

        >>> x = np.linspace(min(bins), max(bins), 10000)
        >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
        ...        / (x * sigma * np.sqrt(2 * np.pi)))

        >>> plt.plot(x, pdf, color='r', linewidth=2)
        >>> plt.show()

        okappaidxBoxMuller2dimRandomState._rand_uint32numpyt_datacntRandomState.dirichletRandomState.logistic (line 4052)kappan <= 0fplam value too large.RandomState.standard_normalneg�+�,�,�
        shuffle(x)

        Modify a sequence in-place by shuffling its contents.

        Parameters
        ----------
        x : array_like
            The array or list to be shuffled.

        Returns
        -------
        None

        Examples
        --------
        >>> arr = np.arange(10)
        >>> mkl_random.shuffle(arr)
        >>> arr
        [1 7 5 2 9 4 3 6 0 8]

        This function only shuffles the array along the first index of a
        multi-dimensional array:

        >>> arr = np.arange(9).reshape((3, 3))
        >>> mkl_random.shuffle(arr)
        >>> arr
        array([[3, 4, 5],
               [6, 7, 8],
               [0, 1, 2]])

        odfdencovsize is not compatible with inputslocInversionastypeallRandomState.tomaxintThe argument to set_state must be a list of 2 elementsget_stateoshapedfRandomState.__reduce__alias_dictoanegative_binomialmulti2chalpha_datasumRandomState.multinormal_choleskyalphalowbndvnsampleRandomState.randint (line 1686)RandomState._rand_int64RandomState.multinomial (line 5543)marrpvalsctypes.RandomState.geometric (line 4935)nskipsfnlogical_or*NON_DETERMINISTICparetogcFewer non-zero entries in p than sizeRandomState.frv_long
        normal(loc=0.0, scale=1.0, size=None, method='ICDF')

        Draw random samples from a normal (Gaussian) distribution.

        The probability density function of the normal distribution, first
        derived by De Moivre and 200 years later by both Gauss and Laplace
        independently [2]_, is often called the bell curve because of
        its characteristic shape (see the example below).

        The normal distributions occurs often in nature.  For example, it
        describes the commonly occurring distribution of samples influenced
        by a large number of tiny, random disturbances, each with its own
        unique distribution [2]_.

        Parameters
        ----------
        loc : float
            Mean ("centre") of the distribution.
        scale : float
            Standard deviation (spread or "width") of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        See Also
        --------
        scipy.stats.distributions.norm : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Gaussian distribution is

        .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
                         e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },

        where :math:`\mu` is the mean and :math:`\sigma` the standard
        deviation. The square of the standard deviation, :math:`\sigma^2`,
        is called the variance.

        The function has its peak at the mean, and its "spread" increases with
        the standard deviation (the function reaches 0.607 times its maximum at
        :math:`x + \sigma` and :math:`x - \sigma` [2]_).  This implies that
        `numpy.random.normal` is more likely to return samples lying close to
        the mean, rather than those far away.

        References
        ----------
        .. [1] Wikipedia, "Normal distribution",
               http://en.wikipedia.org/wiki/Normal_distribution
        .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
               Random Variables and Random Signal Principles", 4th ed., 2001,
               pp. 51, 51, 125.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, sigma = 0, 0.1 # mean and standard deviation
        >>> s = mkl_random.normal(mu, sigma, 1000)

        Verify the mean and the variance:

        >>> abs(mu - np.mean(s)) < 0.01
        True

        >>> abs(sigma - np.std(s, ddof=1)) < 0.01
        True

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 30, normed=True)
        >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
        ...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
        ...          linewidth=2, color='r')
        >>> plt.show()

        beta
        binomial(n, p, size=None)

        Draw samples from a binomial distribution.

        Samples are drawn from a binomial distribution with specified
        parameters, n trials and p probability of success where
        n an integer >= 0 and p is in the interval [0,1]. (n may be
        input as a float, but it is truncated to an integer in use)

        Parameters
        ----------
        n : float (but truncated to an integer)
                parameter, >= 0.
        p : float
                parameter, >= 0 and <=1.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
                  where the values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.binom : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the binomial distribution is

        .. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N},

        where :math:`n` is the number of trials, :math:`p` is the probability
        of success, and :math:`N` is the number of successes.

        When estimating the standard error of a proportion in a population by
        using a random sample, the normal distribution works well unless the
        product p*n <=5, where p = population proportion estimate, and n =
        number of samples, in which case the binomial distribution is used
        instead. For example, a sample of 15 people shows 4 who are left
        handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
        so the binomial distribution should be used in this case.

        References
        ----------
        .. [1] Dalgaard, Peter, "Introductory Statistics with R",
               Springer-Verlag, 2002.
        .. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
               Fifth Edition, 2002.
        .. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden
               and Quigley, 1972.
        .. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A
               Wolfram Web Resource.
               http://mathworld.wolfram.com/BinomialDistribution.html
        .. [5] Wikipedia, "Binomial-distribution",
               http://en.wikipedia.org/wiki/Binomial_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> n, p = 10, .5  # number of trials, probability of each trial
        >>> s = mkl_random.binomial(n, p, 1000)
        # result of flipping a coin 10 times, tested 1000 times.

        A real world example. A company drills 9 wild-cat oil exploration
        wells, each with an estimated probability of success of 0.1. All nine
        wells fail. What is the probability of that happening?

        Let's do 20,000 trials of the model, and count the number that
        generate zero positive results.

        >>> sum(mkl_random.binomial(9, 0.1, 20000) == 0)/20000.
        # answer = 0.38885, or 38%.

        
        weibull(a, size=None)

        Draw samples from a Weibull distribution.

        Draw samples from a 1-parameter Weibull distribution with the given
        shape parameter `a`.

        .. math:: X = (-ln(U))^{1/a}

        Here, U is drawn from the uniform distribution over (0,1].

        The more common 2-parameter Weibull, including a scale parameter
        :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.

        Parameters
        ----------
        a : float
            Shape of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray

        See Also
        --------
        scipy.stats.distributions.weibull_max
        scipy.stats.distributions.weibull_min
        scipy.stats.distributions.genextreme
        gumbel

        Notes
        -----
        The Weibull (or Type III asymptotic extreme value distribution
        for smallest values, SEV Type III, or Rosin-Rammler
        distribution) is one of a class of Generalized Extreme Value
        (GEV) distributions used in modeling extreme value problems.
        This class includes the Gumbel and Frechet distributions.

        The probability density for the Weibull distribution is

        .. math:: p(x) = \frac{a}
                         {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},

        where :math:`a` is the shape and :math:`\lambda` the scale.

        The function has its peak (the mode) at
        :math:`\lambda(\frac{a-1}{a})^{1/a}`.

        When ``a = 1``, the Weibull distribution reduces to the exponential
        distribution.

        References
        ----------
        .. [1] Waloddi Weibull, Royal Technical University, Stockholm,
               1939 "A Statistical Theory Of The Strength Of Materials",
               Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,
               Generalstabens Litografiska Anstalts Forlag, Stockholm.
        .. [2] Waloddi Weibull, "A Statistical Distribution Function of
               Wide Applicability", Journal Of Applied Mechanics ASME Paper
               1951.
        .. [3] Wikipedia, "Weibull distribution",
               http://en.wikipedia.org/wiki/Weibull_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> a = 5. # shape
        >>> s = mkl_random.weibull(a, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> x = np.arange(1,100.)/50.
        >>> def weib(x,n,a):
        ...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)

        >>> count, bins, ignored = plt.hist(mkl_random.weibull(5.,1000))
        >>> x = np.arange(1,100.)/50.
        >>> scale = count.max()/weib(x, 1., 5.).max()
        >>> plt.plot(x, weib(x, 1., 5.)*scale)
        >>> plt.show()

        atolRandomState.rand (line 2169)triangularsza <= 1.0isnanrv_intlamRandomState.__getstate__oleftreducenewRandomState.standard_cauchylessn < 0methodTImportErrornstate_buffer_sizeThe basic random generator specification {given} is not recognized. "MT19937" will be used insteadnsample < 1safe
        rayleigh(scale=1.0, size=None)

        Draw samples from a Rayleigh distribution.

        The :math:`\chi` and Weibull distributions are generalizations of the
        Rayleigh.

        Parameters
        ----------
        scale : scalar
            Scale, also equals the mode. Should be >= 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Notes
        -----
        The probability density function for the Rayleigh distribution is

        .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}

        The Rayleigh distribution would arise, for example, if the East
        and North components of the wind velocity had identical zero-mean
        Gaussian distributions.  Then the wind speed would have a Rayleigh
        distribution.

        References
        ----------
        .. [1] Brighton Webs Ltd., "Rayleigh Distribution,"
               http://www.brighton-webs.co.uk/distributions/rayleigh.asp
        .. [2] Wikipedia, "Rayleigh distribution"
               http://en.wikipedia.org/wiki/Rayleigh_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram

        >>> values = hist(mkl_random.rayleigh(3, 100000), bins=200, normed=True)

        Wave heights tend to follow a Rayleigh distribution. If the mean wave
        height is 1 meter, what fraction of waves are likely to be larger than 3
        meters?

        >>> meanvalue = 1
        >>> modevalue = np.sqrt(2 / np.pi) * meanvalue
        >>> s = mkl_random.rayleigh(modevalue, 1000000)

        The percentage of waves larger than 3 meters is:

        >>> 100.*sum(s>3)/1000000.
        0.087300000000000003

        empty_like']p is nanreshape
        power(a, size=None)

        Draws samples in [0, 1] from a power distribution with positive
        exponent a - 1.

        Also known as the power function distribution.

        Parameters
        ----------
        a : float
            parameter, > 0
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The returned samples lie in [0, 1].

        Raises
        ------
        ValueError
            If a < 1.

        Notes
        -----
        The probability density function is

        .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.

        The power function distribution is just the inverse of the Pareto
        distribution. It may also be seen as a special case of the Beta
        distribution.

        It is used, for example, in modeling the over-reporting of insurance
        claims.

        References
        ----------
        .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions
               in economics and actuarial sciences", Wiley, 2003.
        .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:
               Dataplot Reference Manual, Volume 2: Let Subcommands and Library
               Functions", National Institute of Standards and Technology
               Handbook Series, June 2003.
               http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

        Examples
        --------
        Draw samples from the distribution:

        >>> a = 5. # shape
        >>> samples = 1000
        >>> s = mkl_random.power(a, samples)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, bins=30)
        >>> x = np.linspace(0, 1, 100)
        >>> y = a*x**(a-1.)
        >>> normed_y = samples*np.diff(bins)[0]*y
        >>> plt.plot(x, normed_y)
        >>> plt.show()

        Compare the power function distribution to the inverse of the Pareto.

        >>> from scipy import stats
        >>> rvs = mkl_random.power(5, 1000000)
        >>> rvsp = mkl_random.pareto(5, 1000000)
        >>> xx = np.linspace(0,1,100)
        >>> powpdf = stats.powerlaw.pdf(xx,5)

        >>> plt.figure()
        >>> plt.hist(rvs, bins=50, normed=True)
        >>> plt.plot(xx,powpdf,'r-')
        >>> plt.title('mkl_random.power(5)')

        >>> plt.figure()
        >>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
        >>> plt.plot(xx,powpdf,'r-')
        >>> plt.title('inverse of 1 + mkl_random.pareto(5)')

        >>> plt.figure()
        >>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
        >>> plt.plot(xx,powpdf,'r-')
        >>> plt.title('inverse of stats.pareto(5)')

        errRandomState.tomaxint (line 1375)_randfdfnumbufRandomState.pareto
        gamma(shape, scale=1.0, size=None)

        Draw samples from a Gamma distribution.

        Samples are drawn from a Gamma distribution with specified parameters,
        `shape` (sometimes designated "k") and `scale` (sometimes designated
        "theta"), where both parameters are > 0.

        Parameters
        ----------
        shape : scalar > 0
            The shape of the gamma distribution.
        scale : scalar > 0, optional
            The scale of the gamma distribution.  Default is equal to 1.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray, float
            Returns one sample unless `size` parameter is specified.

        See Also
        --------
        scipy.stats.distributions.gamma : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Gamma distribution is

        .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},

        where :math:`k` is the shape and :math:`\theta` the scale,
        and :math:`\Gamma` is the Gamma function.

        The Gamma distribution is often used to model the times to failure of
        electronic components, and arises naturally in processes for which the
        waiting times between Poisson distributed events are relevant.

        References
        ----------
        .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
               Wolfram Web Resource.
               http://mathworld.wolfram.com/GammaDistribution.html
        .. [2] Wikipedia, "Gamma-distribution",
               http://en.wikipedia.org/wiki/Gamma-distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> shape, scale = 2., 2. # mean and dispersion
        >>> s = mkl_random.gamma(shape, scale, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> import scipy.special as sps
        >>> count, bins, ignored = plt.hist(s, 50, normed=True)
        >>> y = bins**(shape-1)*(np.exp(-bins/scale) /
        ...                      (sps.gamma(shape)*scale**shape))
        >>> plt.plot(bins, y, linewidth=2, color='r')
        >>> plt.show()

        geometricfshapephighbndRandomState.laplace (line 3827)algorithm_nameRandomState.multivariate_normal (line 5232)pix__enter__osigmaanyval_arr__setstate__RandomState.shuffle
        random_sample(size=None)

        Return random floats in the half-open interval [0.0, 1.0).

        Results are from the "continuous uniform" distribution over the
        stated interval.  To sample :math:`Unif[a, b), b > a` multiply
        the output of `random_sample` by `(b-a)` and add `a`::

          (b - a) * random_sample() + a

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : float or ndarray of floats
            Array of random floats of shape `size` (unless ``size=None``, in which
            case a single float is returned).

        Examples
        --------
        >>> mkl_random.random_sample()
        0.47108547995356098
        >>> type(mkl_random.random_sample())
        <type 'float'>
        >>> mkl_random.random_sample((5,))
        array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428])

        Three-by-two array of random numbers from [-5, 0):

        >>> 5 * mkl_random.random_sample((3, 2)) - 5
        array([[-3.99149989, -0.52338984],
               [-2.99091858, -0.79479508],
               [-1.23204345, -1.75224494]])

        getbytestringrng__main__sigmaodfnumfRandomState._rand_int32RandomState.standard_cauchy (line 3236)kappa < 0uint64_RandomState.permutation (line 5863)brng
        poisson(lam=1.0, size=None, method='POISNORM')

        Draw samples from a Poisson distribution.

        The Poisson distribution is the limit of the binomial distribution
        for large N.

        Parameters
        ----------
        lam : float or sequence of float
            Expectation of interval, should be >= 0. A sequence of expectation
            intervals must be broadcastable over the requested size.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'POISNORM, 'PTPE', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.POISNORM, mkl_random.PTPE

        Returns
        -------
        samples : ndarray or scalar
            The drawn samples, of shape *size*, if it was provided.

        Notes
        -----
        The Poisson distribution

        .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}

        For events with an expected separation :math:`\lambda` the Poisson
        distribution :math:`f(k; \lambda)` describes the probability of
        :math:`k` events occurring within the observed
        interval :math:`\lambda`.

        Because the output is limited to the range of the C long type, a
        ValueError is raised when `lam` is within 10 sigma of the maximum
        representable value.

        References
        ----------
        .. [1] Weisstein, Eric W. "Poisson Distribution."
               From MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/PoissonDistribution.html
        .. [2] Wikipedia, "Poisson distribution",
               http://en.wikipedia.org/wiki/Poisson_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> import numpy as np
        >>> s = mkl_random.poisson(5, 10000)

        Display histogram of the sample:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 14, normed=True)
        >>> plt.show()

        Draw each 100 values for lambda 100 and 500:

        >>> s = mkl_random.poisson(lam=(100., 500.), size=(100, 2))

        multinormal_cholesky
        permutation(x)

        Randomly permute a sequence, or return a permuted range.

        If `x` is a multi-dimensional array, it is only shuffled along its
        first index.

        Parameters
        ----------
        x : int or array_like
            If `x` is an integer, randomly permute ``np.arange(x)``.
            If `x` is an array, make a copy and shuffle the elements
            randomly.

        Returns
        -------
        out : ndarray
            Permuted sequence or array range.

        Examples
        --------
        >>> mkl_random.permutation(10)
        array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6])

        >>> mkl_random.permutation([1, 4, 9, 12, 15])
        array([15,  1,  9,  4, 12])

        >>> arr = np.arange(9).reshape((3, 3))
        >>> mkl_random.permutation(arr)
        array([[6, 7, 8],
               [0, 1, 2],
               [3, 4, 5]])

        
        multivariate_normal(mean, cov[, size])

        Draw random samples from a multivariate normal distribution.

        The multivariate normal, multinormal or Gaussian distribution is a
        generalization of the one-dimensional normal distribution to higher
        dimensions.  Such a distribution is specified by its mean and
        covariance matrix.  These parameters are analogous to the mean
        (average or "center") and variance (standard deviation, or "width,"
        squared) of the one-dimensional normal distribution.

        Parameters
        ----------
        mean : 1-D array_like, of length N
            Mean of the N-dimensional distribution.
        cov : 2-D array_like, of shape (N, N)
            Covariance matrix of the distribution. It must be symmetric and
            positive-semidefinite for proper sampling.
        size : int or tuple of ints, optional
            Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
            generated, and packed in an `m`-by-`n`-by-`k` arrangement.  Because
            each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
            If no shape is specified, a single (`N`-D) sample is returned.

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        Notes
        -----
        The mean is a coordinate in N-dimensional space, which represents the
        location where samples are most likely to be generated.  This is
        analogous to the peak of the bell curve for the one-dimensional or
        univariate normal distribution.

        Covariance indicates the level to which two variables vary together.
        From the multivariate normal distribution, we draw N-dimensional
        samples, :math:`X = [x_1, x_2, ... x_N]`.  The covariance matrix
        element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
        The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
        "spread").

        Instead of specifying the full covariance matrix, popular
        approximations include:

          - Spherical covariance (*cov* is a multiple of the identity matrix)
          - Diagonal covariance (*cov* has non-negative elements, and only on
            the diagonal)

        This geometrical property can be seen in two dimensions by plotting
        generated data-points:

        >>> mean = [0, 0]
        >>> cov = [[1, 0], [0, 100]]  # diagonal covariance

        Diagonal covariance means that points are oriented along x or y-axis:

        >>> import matplotlib.pyplot as plt
        >>> x, y = mkl_random.multivariate_normal(mean, cov, 5000).T
        >>> plt.plot(x, y, 'x')
        >>> plt.axis('equal')
        >>> plt.show()

        Note that the covariance matrix must be positive semidefinite (a.k.a.
        nonnegative-definite). Otherwise, the behavior of this method is
        undefined and backwards compatibility is not guaranteed.

        References
        ----------
        .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
               Processes," 3rd ed., New York: McGraw-Hill, 1991.
        .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
               Classification," 2nd ed., New York: Wiley, 2001.

        Examples
        --------
        >>> mean = (1, 2)
        >>> cov = [[1, 0], [0, 1]]
        >>> x = mkl_random.multivariate_normal(mean, cov, (3, 3))
        >>> x.shape
        (3, 3, 2)

        The following is probably true, given that 0.6 is roughly twice the
        standard deviation:

        >>> list((x[0,0,:] - mean) < 0.6)
        [True, True]

        
        pareto(a, size=None)

        Draw samples from a Pareto II or Lomax distribution with
        specified shape.

        The Lomax or Pareto II distribution is a shifted Pareto
        distribution. The classical Pareto distribution can be
        obtained from the Lomax distribution by adding 1 and
        multiplying by the scale parameter ``m`` (see Notes).  The
        smallest value of the Lomax distribution is zero while for the
        classical Pareto distribution it is ``mu``, where the standard
        Pareto distribution has location ``mu = 1``.  Lomax can also
        be considered as a simplified version of the Generalized
        Pareto distribution (available in SciPy), with the scale set
        to one and the location set to zero.

        The Pareto distribution must be greater than zero, and is
        unbounded above.  It is also known as the "80-20 rule".  In
        this distribution, 80 percent of the weights are in the lowest
        20 percent of the range, while the other 20 percent fill the
        remaining 80 percent of the range.

        Parameters
        ----------
        shape : float, > 0.
            Shape of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        See Also
        --------
        scipy.stats.distributions.lomax.pdf : probability density function,
            distribution or cumulative density function, etc.
        scipy.stats.distributions.genpareto.pdf : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Pareto distribution is

        .. math:: p(x) = \frac{a m^a}{(1+x)^{a+1}}

        where :math:`a` is the shape and :math:`m` the scale.

        The Pareto distribution, named after the Italian economist
        Vilfredo Pareto, is a power law probability distribution
        useful in many real world problems.  Outside the field of
        economics it is generally referred to as the Bradford
        distribution. Pareto developed the distribution to describe
        the distribution of wealth in an economy.  It has also found
        use in insurance, web page access statistics, oil field sizes,
        and many other problems, including the download frequency for
        projects in Sourceforge [1]_.  It is one of the so-called
        "fat-tailed" distributions.


        References
        ----------
        .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
               Sourceforge projects.
        .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
        .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
               Values, Birkhauser Verlag, Basel, pp 23-30.
        .. [4] Wikipedia, "Pareto distribution",
               http://en.wikipedia.org/wiki/Pareto_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> a, m = 3., 2.  # shape and mode
        >>> s = (mkl_random.pareto(a, 1000) + 1) * m

        Display the histogram of the samples, along with the probability
        density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, _ = plt.hist(s, 100, normed=True)
        >>> fit = a*m**a / bins**(a+1)
        >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
        >>> plt.show()

        shape <= 0btp >= 1.0fleftu_datafbval_data
        skipahead(nskips)

        Initializes the current state generator using skip-ahead method,
        if supported for the basic random pseudo-random number generation
        algorithm.
        xfinal_shapeaccintegerRandomState.poweruint32onRandomState.random_integers (line 2271)asarrayself
        exponential(scale=1.0, size=None)

        Draw samples from an exponential distribution.

        Its probability density function is

        .. math:: f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),

        for ``x > 0`` and 0 elsewhere. :math:`\beta` is the scale parameter,
        which is the inverse of the rate parameter :math:`\lambda = 1/\beta`.
        The rate parameter is an alternative, widely used parameterization
        of the exponential distribution [3]_.

        The exponential distribution is a continuous analogue of the
        geometric distribution.  It describes many common situations, such as
        the size of raindrops measured over many rainstorms [1]_, or the time
        between page requests to Wikipedia [2]_.

        Parameters
        ----------
        scale : float
            The scale parameter, :math:`\beta = 1/\lambda`.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        References
        ----------
        .. [1] Peyton Z. Peebles Jr., "Probability, Random Variables and
               Random Signal Principles", 4th ed, 2001, p. 57.
        .. [2] "Poisson Process", Wikipedia,
               http://en.wikipedia.org/wiki/Poisson_process
        .. [3] "Exponential Distribution, Wikipedia,
               http://en.wikipedia.org/wiki/Exponential_distribution

        
        bytes(length)

        Return random bytes.

        Parameters
        ----------
        length : int
            Number of random bytes.

        Returns
        -------
        out : str
            String of length `length`.

        Examples
        --------
        >>> mkl_random.bytes(10)
        ' eh\x85\x022SZ\xbf\xa4' #random

        nbad < 0sigma <= 0RandomState.logseries (line 5134)
        laplace(loc=0.0, scale=1.0, size=None)

        Draw samples from the Laplace or double exponential distribution with
        specified location (or mean) and scale (decay).

        The Laplace distribution is similar to the Gaussian/normal distribution,
        but is sharper at the peak and has fatter tails. It represents the
        difference between two independent, identically distributed exponential
        random variables.

        Parameters
        ----------
        loc : float, optional
            The position, :math:`\mu`, of the distribution peak.
        scale : float, optional
            :math:`\lambda`, the exponential decay.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or float

        Notes
        -----
        It has the probability density function

        .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda}
                                       \exp\left(-\frac{|x - \mu|}{\lambda}\right).

        The first law of Laplace, from 1774, states that the frequency
        of an error can be expressed as an exponential function of the
        absolute magnitude of the error, which leads to the Laplace
        distribution. For many problems in economics and health
        sciences, this distribution seems to model the data better
        than the standard Gaussian distribution.

        References
        ----------
        .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
               Mathematical Functions with Formulas, Graphs, and Mathematical
               Tables, 9th printing," New York: Dover, 1972.
        .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and
               Generalizations, " Birkhauser, 2001.
        .. [3] Weisstein, Eric W. "Laplace Distribution."
               From MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/LaplaceDistribution.html
        .. [4] Wikipedia, "Laplace Distribution",
               http://en.wikipedia.org/wiki/Laplace_distribution

        Examples
        --------
        Draw samples from the distribution

        >>> loc, scale = 0., 1.
        >>> s = mkl_random.laplace(loc, scale, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 30, normed=True)
        >>> x = np.arange(-8., 8., .01)
        >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
        >>> plt.plot(x, pdf)

        Plot Gaussian for comparison:

        >>> g = (1/(scale * np.sqrt(2 * np.pi)) *
        ...      np.exp(-(x - loc)**2 / (2 * scale**2)))
        >>> plt.plot(x,g)

        
        negative_binomial(n, p, size=None)

        Draw samples from a negative binomial distribution.

        Samples are drawn from a negative binomial distribution with specified
        parameters, `n` trials and `p` probability of success where `n` is an
        integer > 0 and `p` is in the interval [0, 1].

        Parameters
        ----------
        n : int
            Parameter, > 0.
        p : float
            Parameter, >= 0 and <=1.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : int or ndarray of ints
            Drawn samples.

        Notes
        -----
        The probability density for the negative binomial distribution is

        .. math:: P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N},

        where :math:`n-1` is the number of successes, :math:`p` is the
        probability of success, and :math:`N+n-1` is the number of trials.
        The negative binomial distribution gives the probability of n-1
        successes and N failures in N+n-1 trials, and success on the (N+n)th
        trial.

        If one throws a die repeatedly until the third time a "1" appears,
        then the probability distribution of the number of non-"1"s that
        appear before the third "1" is a negative binomial distribution.

        References
        ----------
        .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
               MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/NegativeBinomialDistribution.html
        .. [2] Wikipedia, "Negative binomial distribution",
               http://en.wikipedia.org/wiki/Negative_binomial_distribution

        Examples
        --------
        Draw samples from the distribution:

        A real world example. A company drills wild-cat oil
        exploration wells, each with an estimated probability of
        success of 0.1.  What is the probability of having one success
        for each successive well, that is what is the probability of a
        single success after drilling 5 wells, after 6 wells, etc.?

        >>> s = mkl_random.negative_binomial(1, 0.1, 100000)
        >>> for i in range(1, 11):
        ...    probability = sum(s<i) / 100000.
        ...    print i, "wells drilled, probability of one success =", probability

        mu_rand_int16dna must be greater than 0
        noncentral_chisquare(df, nonc, size=None)

        Draw samples from a noncentral chi-square distribution.

        The noncentral :math:`\chi^2` distribution is a generalisation of
        the :math:`\chi^2` distribution.

        Parameters
        ----------
        df : int
            Degrees of freedom, should be > 0 as of Numpy 1.10,
            should be > 1 for earlier versions.
        nonc : float
            Non-centrality, should be non-negative.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Notes
        -----
        The probability density function for the noncentral Chi-square
        distribution is

        .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0}
                               \frac{e^{-nonc/2}(nonc/2)^{i}}{i!}
                               \P_{Y_{df+2i}}(x),

        where :math:`Y_{q}` is the Chi-square with q degrees of freedom.

        In Delhi (2007), it is noted that the noncentral chi-square is
        useful in bombing and coverage problems, the probability of
        killing the point target given by the noncentral chi-squared
        distribution.

        References
        ----------
        .. [1] Delhi, M.S. Holla, "On a noncentral chi-square distribution in
               the analysis of weapon systems effectiveness", Metrika,
               Volume 15, Number 1 / December, 1970.
        .. [2] Wikipedia, "Noncentral chi-square distribution"
               http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram

        >>> import matplotlib.pyplot as plt
        >>> values = plt.hist(mkl_random.noncentral_chisquare(3, 20, 100000),
        ...                   bins=200, normed=True)
        >>> plt.show()

        Draw values from a noncentral chisquare with very small noncentrality,
        and compare to a chisquare.

        >>> plt.figure()
        >>> values = plt.hist(mkl_random.noncentral_chisquare(3, .0000001, 100000),
        ...                   bins=np.arange(0., 25, .1), normed=True)
        >>> values2 = plt.hist(mkl_random.chisquare(3, 100000),
        ...                    bins=np.arange(0., 25, .1), normed=True)
        >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
        >>> plt.show()

        Demonstrate how large values of non-centrality lead to a more symmetric
        distribution.

        >>> plt.figure()
        >>> values = plt.hist(mkl_random.noncentral_chisquare(3, 20, 100000),
        ...                   bins=200, normed=True)
        >>> plt.show()

        noncentral_fsqrthighRandomState.vonmises�+�_initializingARS5
        _rand_uint8(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        objRandomState.weibullposbrng_iddfdendf <= 0Skip-ahead method of stream initialization is not supported for flam
        chisquare(df, size=None)

        Draw samples from a chi-square distribution.

        When `df` independent random variables, each with standard normal
        distributions (mean 0, variance 1), are squared and summed, the
        resulting distribution is chi-square (see Notes).  This distribution
        is often used in hypothesis testing.

        Parameters
        ----------
        df : int
             Number of degrees of freedom.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        output : ndarray
            Samples drawn from the distribution, packed in a `size`-shaped
            array.

        Raises
        ------
        ValueError
            When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
            is given.

        Notes
        -----
        The variable obtained by summing the squares of `df` independent,
        standard normally distributed random variables:

        .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i

        is chi-square distributed, denoted

        .. math:: Q \sim \chi^2_k.

        The probability density function of the chi-squared distribution is

        .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
                         x^{k/2 - 1} e^{-x/2},

        where :math:`\Gamma` is the gamma function,

        .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.

        References
        ----------
        .. [1] NIST "Engineering Statistics Handbook"
               http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

        Examples
        --------
        >>> mkl_random.chisquare(2,4)
        array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272])

        
        chisquare(df, size=None)

        Draw samples from a chi-square distribution.

        When `df` independent random variables, each with standard normal
        distributions (mean 0, variance 1), are squared and summed, the
        resulting distribution is chi-square (see Notes).  This distribution
        is often used in hypothesis testing.

        Parameters
        ----------
        df : int
             Number of degrees of freedom.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        output : ndarray
            Samples drawn from the distribution, packed in a `size`-shaped
            array.

        Raises
        ------
        ValueError
            When `df` <= 0 or when an inappropriate `size` (e.g. ``size=-1``)
            is given.

        Notes
        -----
        The variable obtained by summing the squares of `df` independent,
        standard normally distributed random variables:

        .. math:: Q = \sum_{i=0}^{\mathtt{df}} X^2_i

        is chi-square distributed, denoted

        .. math:: Q \sim \chi^2_k.

        The probability density function of the chi-squared distribution is

        .. math:: p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
                         x^{k/2 - 1} e^{-x/2},

        where :math:`\Gamma` is the gamma function,

        .. math:: \Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.

        References
        ----------
        .. [1] NIST "Engineering Statistics Handbook"
               http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

        Examples
        --------
        >>> mkl_random.chisquare(2,4)
        array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272])

        
        gumbel(loc=0.0, scale=1.0, size=None)

        Draw samples from a Gumbel distribution.

        Draw samples from a Gumbel distribution with specified location and
        scale.  For more information on the Gumbel distribution, see
        Notes and References below.

        Parameters
        ----------
        loc : float
            The location of the mode of the distribution.
        scale : float
            The scale parameter of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar

        See Also
        --------
        scipy.stats.gumbel_l
        scipy.stats.gumbel_r
        scipy.stats.genextreme
        weibull

        Notes
        -----
        The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
        Value Type I) distribution is one of a class of Generalized Extreme
        Value (GEV) distributions used in modeling extreme value problems.
        The Gumbel is a special case of the Extreme Value Type I distribution
        for maximums from distributions with "exponential-like" tails.

        The probability density for the Gumbel distribution is

        .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
                  \beta}},

        where :math:`\mu` is the mode, a location parameter, and
        :math:`\beta` is the scale parameter.

        The Gumbel (named for German mathematician Emil Julius Gumbel) was used
        very early in the hydrology literature, for modeling the occurrence of
        flood events. It is also used for modeling maximum wind speed and
        rainfall rates.  It is a "fat-tailed" distribution - the probability of
        an event in the tail of the distribution is larger than if one used a
        Gaussian, hence the surprisingly frequent occurrence of 100-year
        floods. Floods were initially modeled as a Gaussian process, which
        underestimated the frequency of extreme events.

        It is one of a class of extreme value distributions, the Generalized
        Extreme Value (GEV) distributions, which also includes the Weibull and
        Frechet.

        The function has a mean of :math:`\mu + 0.57721\beta` and a variance
        of :math:`\frac{\pi^2}{6}\beta^2`.

        References
        ----------
        .. [1] Gumbel, E. J., "Statistics of Extremes,"
               New York: Columbia University Press, 1958.
        .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
               Values from Insurance, Finance, Hydrology and Other Fields,"
               Basel: Birkhauser Verlag, 2001.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, beta = 0, 0.1 # location and scale
        >>> s = mkl_random.gumbel(mu, beta, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 30, normed=True)
        >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
        ...          * np.exp( -np.exp( -(bins - mu) /beta) ),
        ...          linewidth=2, color='r')
        >>> plt.show()

        Show how an extreme value distribution can arise from a Gaussian process
        and compare to a Gaussian:

        >>> means = []
        >>> maxima = []
        >>> for i in range(0,1000) :
        ...    a = mkl_random.normal(mu, beta, 1000)
        ...    means.append(a.mean())
        ...    maxima.append(a.max())
        >>> count, bins, ignored = plt.hist(maxima, 30, normed=True)
        >>> beta = np.std(maxima) * np.sqrt(6) / np.pi
        >>> mu = np.mean(maxima) - 0.57721*beta
        >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
        ...          * np.exp(-np.exp(-(bins - mu)/beta)),
        ...          linewidth=2, color='r')
        >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
        ...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
        ...          linewidth=2, color='g')
        >>> plt.show()

        Leap-frog method of stream initialization is not supported for probabilities do not sum to 1mean <= 0fkappamaxrandint
        multinomial(n, pvals, size=None, method='ICDF')

        Draw samples from a multinomial distribution.

        The multinomial distribution is a multivariate generalisation of the
        binomial distribution.  Take an experiment with one of ``p``
        possible outcomes.  An example of such an experiment is throwing a dice,
        where the outcome can be 1 through 6.  Each sample drawn from the
        distribution represents `n` such experiments.  Its values,
        ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the
        outcome was ``i``.

        Parameters
        ----------
        n : int
            Number of experiments.
        pvals : sequence of floats, length p
            Probabilities of each of the ``p`` different outcomes.  These
            should sum to 1 (however, the last element is always assumed to
            account for the remaining probability, as long as
            ``sum(pvals[:-1]) <= 1)``.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        Examples
        --------
        Throw a dice 20 times:

        >>> mkl_random.multinomial(20, [1/6.]*6, size=1)
        array([[4, 1, 7, 5, 2, 1]])

        It landed 4 times on 1, once on 2, etc.

        Now, throw the dice 20 times, and 20 times again:

        >>> mkl_random.multinomial(20, [1/6.]*6, size=2)
        array([[3, 4, 3, 3, 4, 3],
               [2, 4, 3, 4, 0, 7]])

        For the first run, we threw 3 times 1, 4 times 2, etc.  For the second,
        we threw 2 times 1, 4 times 2, etc.

        A loaded die is more likely to land on number 6:

        >>> mkl_random.multinomial(100, [1/7.]*5 + [2./7.])
        array([11, 16, 14, 17, 16, 26])

        The probability inputs should be normalized. As an implementation
        detail, the value of the last entry is ignored and assumed to take
        up any leftover probability mass, but this should not be relied on.
        A biased coin which has twice as much weight on one side as on the
        other should be sampled like so:

        >>> mkl_random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
        array([38, 62])

        not like:

        >>> mkl_random.multinomial(100, [1.0, 2.0])  # WRONG
        array([100,   0])

        ValueError?RandomState.normal (line 2408)gammaRandomState.laplaceint64
        standard_gamma(shape, size=None)

        Draw samples from a standard Gamma distribution.

        Samples are drawn from a Gamma distribution with specified parameters,
        shape (sometimes designated "k") and scale=1.

        Parameters
        ----------
        shape : float
            Parameter, should be > 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The drawn samples.

        See Also
        --------
        scipy.stats.distributions.gamma : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Gamma distribution is

        .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},

        where :math:`k` is the shape and :math:`\theta` the scale,
        and :math:`\Gamma` is the Gamma function.

        The Gamma distribution is often used to model the times to failure of
        electronic components, and arises naturally in processes for which the
        waiting times between Poisson distributed events are relevant.

        References
        ----------
        .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
               Wolfram Web Resource.
               http://mathworld.wolfram.com/GammaDistribution.html
        .. [2] Wikipedia, "Gamma-distribution",
               http://en.wikipedia.org/wiki/Gamma-distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> shape, scale = 2., 1. # mean and width
        >>> s = mkl_random.standard_gamma(shape, 1000000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> import scipy.special as sps
        >>> count, bins, ignored = plt.hist(s, 50, normed=True)
        >>> y = bins**(shape-1) * ((np.exp(-bins/scale))/ \
        ...                       (sps.gamma(shape) * scale**shape))
        >>> plt.plot(bins, y, linewidth=2, color='r')
        >>> plt.show()

        RandomState.choice (line 1888)NONDETERMfscalekeymode > rightp < 0.0shapefloat64numpy.linalg
        vonmises(mu, kappa, size=None)

        Draw samples from a von Mises distribution.

        Samples are drawn from a von Mises distribution with specified mode
        (mu) and dispersion (kappa), on the interval [-pi, pi].

        The von Mises distribution (also known as the circular normal
        distribution) is a continuous probability distribution on the unit
        circle.  It may be thought of as the circular analogue of the normal
        distribution.

        Parameters
        ----------
        mu : float
            Mode ("center") of the distribution.
        kappa : float
            Dispersion of the distribution, has to be >=0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : scalar or ndarray
            The returned samples, which are in the interval [-pi, pi].

        See Also
        --------
        scipy.stats.distributions.vonmises : probability density function,
            distribution, or cumulative density function, etc.

        Notes
        -----
        The probability density for the von Mises distribution is

        .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},

        where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
        and :math:`I_0(\kappa)` is the modified Bessel function of order 0.

        The von Mises is named for Richard Edler von Mises, who was born in
        Austria-Hungary, in what is now the Ukraine.  He fled to the United
        States in 1939 and became a professor at Harvard.  He worked in
        probability theory, aerodynamics, fluid mechanics, and philosophy of
        science.

        References
        ----------
        .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
               Mathematical Functions with Formulas, Graphs, and Mathematical
               Tables, 9th printing," New York: Dover, 1972.
        .. [2] von Mises, R., "Mathematical Theory of Probability
               and Statistics", New York: Academic Press, 1964.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, kappa = 0.0, 4.0 # mean and dispersion
        >>> s = mkl_random.vonmises(mu, kappa, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> from scipy.special import i0
        >>> plt.hist(s, 50, normed=True)
        >>> x = np.linspace(-np.pi, np.pi, num=51)
        >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))
        >>> plt.plot(x, y, linewidth=2, color='r')
        >>> plt.show()

        _brng_dicti_rand_uint32
        _rand_uint64(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        range_brng_dict_stream_maxRandomState.standard_exponential (line 2644)MT19937wald
        zipf(a, size=None)

        Draw samples from a Zipf distribution.

        Samples are drawn from a Zipf distribution with specified parameter
        `a` > 1.

        The Zipf distribution (also known as the zeta distribution) is a
        continuous probability distribution that satisfies Zipf's law: the
        frequency of an item is inversely proportional to its rank in a
        frequency table.

        Parameters
        ----------
        a : float > 1
            Distribution parameter.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : scalar or ndarray
            The returned samples are greater than or equal to one.

        See Also
        --------
        scipy.stats.distributions.zipf : probability density function,
            distribution, or cumulative density function, etc.

        Notes
        -----
        The probability density for the Zipf distribution is

        .. math:: p(x) = \frac{x^{-a}}{\zeta(a)},

        where :math:`\zeta` is the Riemann Zeta function.

        It is named for the American linguist George Kingsley Zipf, who noted
        that the frequency of any word in a sample of a language is inversely
        proportional to its rank in the frequency table.

        References
        ----------
        .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative
               Frequency in Language," Cambridge, MA: Harvard Univ. Press,
               1932.

        Examples
        --------
        Draw samples from the distribution:

        >>> a = 2. # parameter
        >>> s = mkl_random.zipf(a, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> import scipy.special as sps
        Truncate s values at 50 so plot is interesting
        >>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
        >>> x = np.arange(1., 50.)
        >>> y = x**(-a)/sps.zetac(a)
        >>> plt.plot(x, y/max(y), linewidth=2, color='r')
        >>> plt.show()

        x_ptr
        shuffle(x)

        Modify a sequence in-place by shuffling its contents.

        Parameters
        ----------
        x : array_like
            The array or list to be shuffled.

        Returns
        -------
        None

        Examples
        --------
        >>> arr = np.arange(10)
        >>> mkl_random.shuffle(arr)
        >>> arr
        [1 7 5 2 9 4 3 6 0 8]

        This function only shuffles the array along the first index of a
        multi-dimensional array:

        >>> arr = np.arange(9).reshape((3, 3))
        >>> mkl_random.shuffle(arr)
        >>> arr
        array([[3, 4, 5],
               [6, 7, 8],
               [0, 1, 2]])

        oscalenp_method_alias_dict_poissonstorage_mode_brng_id_to_name
        gamma(shape, scale=1.0, size=None)

        Draw samples from a Gamma distribution.

        Samples are drawn from a Gamma distribution with specified parameters,
        `shape` (sometimes designated "k") and `scale` (sometimes designated
        "theta"), where both parameters are > 0.

        Parameters
        ----------
        shape : scalar > 0
            The shape of the gamma distribution.
        scale : scalar > 0, optional
            The scale of the gamma distribution.  Default is equal to 1.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray, float
            Returns one sample unless `size` parameter is specified.

        See Also
        --------
        scipy.stats.distributions.gamma : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Gamma distribution is

        .. math:: p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},

        where :math:`k` is the shape and :math:`\theta` the scale,
        and :math:`\Gamma` is the Gamma function.

        The Gamma distribution is often used to model the times to failure of
        electronic components, and arises naturally in processes for which the
        waiting times between Poisson distributed events are relevant.

        References
        ----------
        .. [1] Weisstein, Eric W. "Gamma Distribution." From MathWorld--A
               Wolfram Web Resource.
               http://mathworld.wolfram.com/GammaDistribution.html
        .. [2] Wikipedia, "Gamma-distribution",
               http://en.wikipedia.org/wiki/Gamma-distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> shape, scale = 2., 2. # mean and dispersion
        >>> s = mkl_random.gamma(shape, scale, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> import scipy.special as sps
        >>> count, bins, ignored = plt.hist(s, 50, normed=True)
        >>> y = bins**(shape-1)*(np.exp(-bins/scale) /
        ...                      (sps.gamma(shape)*scale**shape))
        >>> plt.plot(bins, y, linewidth=2, color='r')
        >>> plt.show()

        modeskipaheadmnixp > 1frightSumdfnum <= 1out_shape_from_sizeuppermulti1svdnumpy._core.umath failed to importfa_is_coroutinebrng_namePoisson-Normalweibull
        dirichlet(alpha, size=None)

        Draw samples from the Dirichlet distribution.

        Draw `size` samples of dimension k from a Dirichlet distribution. A
        Dirichlet-distributed random variable can be seen as a multivariate
        generalization of a Beta distribution. Dirichlet pdf is the conjugate
        prior of a multinomial in Bayesian inference.

        Parameters
        ----------
        alpha : array
            Parameter of the distribution (k dimension for sample of
            dimension k).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray,
            The drawn samples, of shape (size, alpha.ndim).

        Notes
        -----
        .. math:: X \approx \prod_{i=1}^{k}{x^{\alpha_i-1}_i}

        Uses the following property for computation: for each dimension,
        draw a random sample y_i from a standard gamma generator of shape
        `alpha_i`, then
        :math:`X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)` is
        Dirichlet distributed.

        References
        ----------
        .. [1] David McKay, "Information Theory, Inference and Learning
               Algorithms," chapter 23,
               http://www.inference.phy.cam.ac.uk/mackay/
        .. [2] Wikipedia, "Dirichlet distribution",
               http://en.wikipedia.org/wiki/Dirichlet_distribution

        Examples
        --------
        Taking an example cited in Wikipedia, this distribution can be used if
        one wanted to cut strings (each of initial length 1.0) into K pieces
        with different lengths, where each piece had, on average, a designated
        average length, but allowing some variation in the relative sizes of
        the pieces.

        >>> s = mkl_random.dirichlet((10, 5, 3), 20).transpose()

        >>> plt.barh(range(20), s[0])
        >>> plt.barh(range(20), s[1], left=s[0], color='g')
        >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
        >>> plt.title("Lengths of Strings")

        
        randn(d0, d1, ..., dn)

        Return a sample (or samples) from the "standard normal" distribution.

        If positive, int_like or int-convertible arguments are provided,
        `randn` generates an array of shape ``(d0, d1, ..., dn)``, filled
        with random floats sampled from a univariate "normal" (Gaussian)
        distribution of mean 0 and variance 1 (if any of the :math:`d_i` are
        floats, they are first converted to integers by truncation). A single
        float randomly sampled from the distribution is returned if no
        argument is provided.

        This is a convenience function.  If you want an interface that takes a
        tuple as the first argument, use `numpy.random.standard_normal` instead.

        Parameters
        ----------
        d0, d1, ..., dn : int, optional
            The dimensions of the returned array, should be all positive.
            If no argument is given a single Python float is returned.

        Returns
        -------
        Z : ndarray or float
            A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from
            the standard normal distribution, or a single such float if
            no parameters were supplied.

        See Also
        --------
        random.standard_normal : Similar, but takes a tuple as its argument.

        Notes
        -----
        For random samples from :math:`N(\mu, \sigma^2)`, use:

        ``sigma * mkl_random.randn(...) + mu``

        Examples
        --------
        >>> mkl_random.randn()
        2.1923875335537315 #random

        Two-by-four array of samples from N(3, 6.25):

        >>> 2.5 * mkl_random.randn(2, 4) + 3
        array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],  #random
               [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]]) #random

        RandomState.noncentral_chisquare
        triangular(left, mode, right, size=None)

        Draw samples from the triangular distribution.

        The triangular distribution is a continuous probability
        distribution with lower limit left, peak at mode, and upper
        limit right. Unlike the other distributions, these parameters
        directly define the shape of the pdf.

        Parameters
        ----------
        left : scalar
            Lower limit.
        mode : scalar
            The value where the peak of the distribution occurs.
            The value should fulfill the condition ``left <= mode <= right``.
        right : scalar
            Upper limit, should be larger than `left`.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The returned samples all lie in the interval [left, right].

        Notes
        -----
        The probability density function for the triangular distribution is

        .. math:: P(x;l, m, r) = \begin{cases}
                  \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\
                  \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\
                  0& \text{otherwise}.
                  \end{cases}

        The triangular distribution is often used in ill-defined
        problems where the underlying distribution is not known, but
        some knowledge of the limits and mode exists. Often it is used
        in simulations.

        References
        ----------
        .. [1] Wikipedia, "Triangular distribution"
               http://en.wikipedia.org/wiki/Triangular_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram:

        >>> import matplotlib.pyplot as plt
        >>> h = plt.hist(mkl_random.triangular(-3, 0, 8, 100000), bins=200,
        ...              normed=True)
        >>> plt.show()

        stridebprobabilities are not non-negativelo__spec__arrayfdfden',RandomState.rayleighRandomState.__setstate__equal
        weibull(a, size=None)

        Draw samples from a Weibull distribution.

        Draw samples from a 1-parameter Weibull distribution with the given
        shape parameter `a`.

        .. math:: X = (-ln(U))^{1/a}

        Here, U is drawn from the uniform distribution over (0,1].

        The more common 2-parameter Weibull, including a scale parameter
        :math:`\lambda` is just :math:`X = \lambda(-ln(U))^{1/a}`.

        Parameters
        ----------
        a : float
            Shape of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray

        See Also
        --------
        scipy.stats.distributions.weibull_max
        scipy.stats.distributions.weibull_min
        scipy.stats.distributions.genextreme
        gumbel

        Notes
        -----
        The Weibull (or Type III asymptotic extreme value distribution
        for smallest values, SEV Type III, or Rosin-Rammler
        distribution) is one of a class of Generalized Extreme Value
        (GEV) distributions used in modeling extreme value problems.
        This class includes the Gumbel and Frechet distributions.

        The probability density for the Weibull distribution is

        .. math:: p(x) = \frac{a}
                         {\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},

        where :math:`a` is the shape and :math:`\lambda` the scale.

        The function has its peak (the mode) at
        :math:`\lambda(\frac{a-1}{a})^{1/a}`.

        When ``a = 1``, the Weibull distribution reduces to the exponential
        distribution.

        References
        ----------
        .. [1] Waloddi Weibull, Royal Technical University, Stockholm,
               1939 "A Statistical Theory Of The Strength Of Materials",
               Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939,
               Generalstabens Litografiska Anstalts Forlag, Stockholm.
        .. [2] Waloddi Weibull, "A Statistical Distribution Function of
               Wide Applicability", Journal Of Applied Mechanics ASME Paper
               1951.
        .. [3] Wikipedia, "Weibull distribution",
               http://en.wikipedia.org/wiki/Weibull_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> a = 5. # shape
        >>> s = mkl_random.weibull(a, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> x = np.arange(1,100.)/50.
        >>> def weib(x,n,a):
        ...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)

        >>> count, bins, ignored = plt.hist(mkl_random.weibull(5.,1000))
        >>> x = np.arange(1,100.)/50.
        >>> scale = count.max()/weib(x, 1., 5.).max()
        >>> plt.plot(x, weib(x, 1., 5.)*scale)
        >>> plt.show()

        ICDFRandomState.logisticRandomState.geometricNumber of balls in each urn should not exceed 2147483647zerosarangelnRandomState.noncentral_fint16cumsumsbinomialnoncstream_idstandard_tUserWarningpower
        standard_normal(size=None, method='ICDF')

        Draw samples from a standard Normal distribution (mean=0, stdev=1).

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        Returns
        -------
        out : float or ndarray
            Drawn samples.

        Examples
        --------
        >>> s = mkl_random.standard_normal(8000)
        >>> s
        array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311, #random
               -0.38672696, -0.4685006 ])                               #random
        >>> s.shape
        (8000,)
        >>> s = mkl_random.standard_normal(size=(3, 4, 2))
        >>> s.shape
        (3, 4, 2)

        bytesb <= 0res_rand_uint64p > 1.0array_long_dataRandomState.triangular (line 4443)arrRandomState.get_statepoissonmean must be 1 dimensionaldAll parameters should not exceed 2147483647nmRandomState._rand_uint16ngooduint16dotThe stream state buffer is corruptedRandomState.random_samplegumbelless_equal
        randn(d0, d1, ..., dn)

        Return a sample (or samples) from the "standard normal" distribution.

        If positive, int_like or int-convertible arguments are provided,
        `randn` generates an array of shape ``(d0, d1, ..., dn)``, filled
        with random floats sampled from a univariate "normal" (Gaussian)
        distribution of mean 0 and variance 1 (if any of the :math:`d_i` are
        floats, they are first converted to integers by truncation). A single
        float randomly sampled from the distribution is returned if no
        argument is provided.

        This is a convenience function.  If you want an interface that takes a
        tuple as the first argument, use `numpy.random.standard_normal` instead.

        Parameters
        ----------
        d0, d1, ..., dn : int, optional
            The dimensions of the returned array, should be all positive.
            If no argument is given a single Python float is returned.

        Returns
        -------
        Z : ndarray or float
            A ``(d0, d1, ..., dn)``-shaped array of floating-point samples from
            the standard normal distribution, or a single such float if
            no parameters were supplied.

        See Also
        --------
        random.standard_normal : Similar, but takes a tuple as its argument.

        Notes
        -----
        For random samples from :math:`N(\mu, \sigma^2)`, use:

        ``sigma * mkl_random.randn(...) + mu``

        Examples
        --------
        >>> mkl_random.randn()
        2.1923875335537315 #random

        Two-by-four array of samples from N(3, 6.25):

        >>> 2.5 * mkl_random.randn(2, 4) + 3
        array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],  #random
               [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]]) #random

        castingdfden <= 0warnings__import__
        poisson(lam=1.0, size=None, method='POISNORM')

        Draw samples from a Poisson distribution.

        The Poisson distribution is the limit of the binomial distribution
        for large N.

        Parameters
        ----------
        lam : float or sequence of float
            Expectation of interval, should be >= 0. A sequence of expectation
            intervals must be broadcastable over the requested size.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'POISNORM, 'PTPE', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.POISNORM, mkl_random.PTPE

        Returns
        -------
        samples : ndarray or scalar
            The drawn samples, of shape *size*, if it was provided.

        Notes
        -----
        The Poisson distribution

        .. math:: f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}

        For events with an expected separation :math:`\lambda` the Poisson
        distribution :math:`f(k; \lambda)` describes the probability of
        :math:`k` events occurring within the observed
        interval :math:`\lambda`.

        Because the output is limited to the range of the C long type, a
        ValueError is raised when `lam` is within 10 sigma of the maximum
        representable value.

        References
        ----------
        .. [1] Weisstein, Eric W. "Poisson Distribution."
               From MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/PoissonDistribution.html
        .. [2] Wikipedia, "Poisson distribution",
               http://en.wikipedia.org/wiki/Poisson_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> import numpy as np
        >>> s = mkl_random.poisson(5, 10000)

        Display histogram of the sample:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 14, normed=True)
        >>> plt.show()

        Draw each 100 values for lambda 100 and 500:

        >>> s = mkl_random.poisson(lam=(100., 500.), size=(100, 2))

        RandomState.wald (line 4358)RandomState.vonmises (line 3405)
        multivariate_normal(mean, ch, size=None, method='ICDF')

        Draw random samples from a multivariate normal distribution.

        The multivariate normal, multinormal or Gaussian distribution is a
        generalization of the one-dimensional normal distribution to higher
        dimensions.  Such a distribution is specified by its mean and
        covariance matrix, specified by its lower triangular Cholesky factor.
        These parameters are analogous to the mean
        (average or "center") and standard deviation, or "width,"
        of the one-dimensional normal distribution.

        Parameters
        ----------
        mean : 1-D array_like, of length N
            Mean of the N-dimensional distribution.
        ch : 2-D array_like, of shape (N, N)
            Cholesky factor of the covariance matrix of the distribution. Only lower-triangular
            part of the matrix is actually used.
        size : int or tuple of ints, optional
            Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
            generated, and packed in an `m`-by-`n`-by-`k` arrangement.  Because
            each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
            If no shape is specified, a single (`N`-D) sample is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        Notes
        -----
        The mean is a coordinate in N-dimensional space, which represents the
        location where samples are most likely to be generated.  This is
        analogous to the peak of the bell curve for the one-dimensional or
        univariate normal distribution.

        Covariance indicates the level to which two variables vary together.
        From the multivariate normal distribution, we draw N-dimensional
        samples, :math:`X = [x_1, x_2, ... x_N]`.  The covariance matrix
        element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
        The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
        "spread").

        Instead of specifying the full covariance matrix, popular
        approximations include:

          - Spherical covariance (*cov* is a multiple of the identity matrix)
          - Diagonal covariance (*cov* has non-negative elements, and only on
            the diagonal)

        This geometrical property can be seen in two dimensions by plotting
        generated data-points:

        >>> mean = [0, 0]
        >>> cov = [[1, 0], [0, 100]]  # diagonal covariance

        Diagonal covariance means that points are oriented along x or y-axis:

        >>> import matplotlib.pyplot as plt
        >>> x, y = mkl_random.multivariate_normal(mean, cov, 5000).T
        >>> plt.plot(x, y, 'x')
        >>> plt.axis('equal')
        >>> plt.show()

        Note that the covariance matrix must be positive semidefinite (a.k.a.
        nonnegative-definite). Otherwise, the behavior of this method is
        undefined and backwards compatibility is not guaranteed.

        References
        ----------
        .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
               Processes," 3rd ed., New York: McGraw-Hill, 1991.
        .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
               Classification," 2nd ed., New York: Wiley, 2001.

        Examples
        --------
        >>> mean = (1, 2)
        >>> cov = [[1, 0], [0, 1]]
        >>> x = mkl_random.multivariate_normal(mean, cov, (3, 3))
        >>> x.shape
        (3, 3, 2)

        The following is probably true, given that 0.6 is roughly twice the
        standard deviation:

        >>> list((x[0,0,:] - mean) < 0.6)
        [True, True]

        smaxBOXMULLER2Parameter alpha is not a vectorRandomState.randint_untyped (line 1767)__class_getitem__take
        standard_cauchy(size=None)

        Draw samples from a standard Cauchy distribution with mode = 0.

        Also known as the Lorentz distribution.

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The drawn samples.

        Notes
        -----
        The probability density function for the full Cauchy distribution is

        .. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
                  (\frac{x-x_0}{\gamma})^2 \bigr] }

        and the Standard Cauchy distribution just sets :math:`x_0=0` and
        :math:`\gamma=1`

        The Cauchy distribution arises in the solution to the driven harmonic
        oscillator problem, and also describes spectral line broadening. It
        also describes the distribution of values at which a line tilted at
        a random angle will cut the x axis.

        When studying hypothesis tests that assume normality, seeing how the
        tests perform on data from a Cauchy distribution is a good indicator of
        their sensitivity to a heavy-tailed distribution, since the Cauchy looks
        very much like a Gaussian distribution, but with heavier tails.

        References
        ----------
        .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy
              Distribution",
              http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
        .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A
              Wolfram Web Resource.
              http://mathworld.wolfram.com/CauchyDistribution.html
        .. [3] Wikipedia, "Cauchy distribution"
              http://en.wikipedia.org/wiki/Cauchy_distribution

        Examples
        --------
        Draw samples and plot the distribution:

        >>> s = mkl_random.standard_cauchy(1000000)
        >>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
        >>> plt.hist(s, bins=100)
        >>> plt.show()

        set_state
        standard_normal(size=None, method='ICDF')

        Draw samples from a standard Normal distribution (mean=0, stdev=1).

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        Returns
        -------
        out : float or ndarray
            Drawn samples.

        Examples
        --------
        >>> s = mkl_random.standard_normal(8000)
        >>> s
        array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311, #random
               -0.38672696, -0.4685006 ])                               #random
        >>> s.shape
        (8000,)
        >>> s = mkl_random.standard_normal(size=(3, 4, 2))
        >>> s.shape
        (3, 4, 2)

        meanisfinitengood + nbad < nsampleRandomState.choicelow >= higholoc
        pareto(a, size=None)

        Draw samples from a Pareto II or Lomax distribution with
        specified shape.

        The Lomax or Pareto II distribution is a shifted Pareto
        distribution. The classical Pareto distribution can be
        obtained from the Lomax distribution by adding 1 and
        multiplying by the scale parameter ``m`` (see Notes).  The
        smallest value of the Lomax distribution is zero while for the
        classical Pareto distribution it is ``mu``, where the standard
        Pareto distribution has location ``mu = 1``.  Lomax can also
        be considered as a simplified version of the Generalized
        Pareto distribution (available in SciPy), with the scale set
        to one and the location set to zero.

        The Pareto distribution must be greater than zero, and is
        unbounded above.  It is also known as the "80-20 rule".  In
        this distribution, 80 percent of the weights are in the lowest
        20 percent of the range, while the other 20 percent fill the
        remaining 80 percent of the range.

        Parameters
        ----------
        shape : float, > 0.
            Shape of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        See Also
        --------
        scipy.stats.distributions.lomax.pdf : probability density function,
            distribution or cumulative density function, etc.
        scipy.stats.distributions.genpareto.pdf : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Pareto distribution is

        .. math:: p(x) = \frac{a m^a}{(1+x)^{a+1}}

        where :math:`a` is the shape and :math:`m` the scale.

        The Pareto distribution, named after the Italian economist
        Vilfredo Pareto, is a power law probability distribution
        useful in many real world problems.  Outside the field of
        economics it is generally referred to as the Bradford
        distribution. Pareto developed the distribution to describe
        the distribution of wealth in an economy.  It has also found
        use in insurance, web page access statistics, oil field sizes,
        and many other problems, including the download frequency for
        projects in Sourceforge [1]_.  It is one of the so-called
        "fat-tailed" distributions.


        References
        ----------
        .. [1] Francis Hunt and Paul Johnson, On the Pareto Distribution of
               Sourceforge projects.
        .. [2] Pareto, V. (1896). Course of Political Economy. Lausanne.
        .. [3] Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme
               Values, Birkhauser Verlag, Basel, pp 23-30.
        .. [4] Wikipedia, "Pareto distribution",
               http://en.wikipedia.org/wiki/Pareto_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> a, m = 3., 2.  # shape and mode
        >>> s = (mkl_random.pareto(a, 1000) + 1) * m

        Display the histogram of the samples, along with the probability
        density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, _ = plt.hist(s, 100, normed=True)
        >>> fit = a*m**a / bins**(a+1)
        >>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
        >>> plt.show()

        
        f(dfnum, dfden, size=None)

        Draw samples from an F distribution.

        Samples are drawn from an F distribution with specified parameters,
        `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
        freedom in denominator), where both parameters should be greater than
        zero.

        The random variate of the F distribution (also known as the
        Fisher distribution) is a continuous probability distribution
        that arises in ANOVA tests, and is the ratio of two chi-square
        variates.

        Parameters
        ----------
        dfnum : float
            Degrees of freedom in numerator. Should be greater than zero.
        dfden : float
            Degrees of freedom in denominator. Should be greater than zero.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            Samples from the Fisher distribution.

        See Also
        --------
        scipy.stats.distributions.f : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The F statistic is used to compare in-group variances to between-group
        variances. Calculating the distribution depends on the sampling, and
        so it is a function of the respective degrees of freedom in the
        problem.  The variable `dfnum` is the number of samples minus one, the
        between-groups degrees of freedom, while `dfden` is the within-groups
        degrees of freedom, the sum of the number of samples in each group
        minus the number of groups.

        References
        ----------
        .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
               Fifth Edition, 2002.
        .. [2] Wikipedia, "F-distribution",
               http://en.wikipedia.org/wiki/F-distribution

        Examples
        --------
        An example from Glantz[1], pp 47-40:

        Two groups, children of diabetics (25 people) and children from people
        without diabetes (25 controls). Fasting blood glucose was measured,
        case group had a mean value of 86.1, controls had a mean value of
        82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
        data consistent with the null hypothesis that the parents diabetic
        status does not affect their children's blood glucose levels?
        Calculating the F statistic from the data gives a value of 36.01.

        Draw samples from the distribution:

        >>> dfnum = 1. # between group degrees of freedom
        >>> dfden = 48. # within groups degrees of freedom
        >>> s = mkl_random.f(dfnum, dfden, 1000)

        The lower bound for the top 1% of the samples is :

        >>> sort(s)[-10]
        7.61988120985

        So there is about a 1% chance that the F statistic will exceed 7.62,
        the measured value is 36, so the null hypothesis is rejected at the 1%
        level.

        MRG32K3AndarrayRandomState.normalRandomState.lognormal (line 4143)__reduce__RandomState.multivariate_normaltarrPTPEcovariance is not positive-semidefinite.axis
        random_integers(low, high=None, size=None)

        Random integers of type np.int between `low` and `high`, inclusive.

        Return random integers of type np.int from the "discrete uniform"
        distribution in the closed interval [`low`, `high`].  If `high` is
        None (the default), then results are from [1, `low`]. The np.int
        type translates to the C long type used by Python 2 for "short"
        integers and its precision is platform dependent.

        This function has been deprecated. Use randint instead.

        .. deprecated:: 1.11.0

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution (unless
            ``high=None``, in which case this parameter is the *highest* such
            integer).
        high : int, optional
            If provided, the largest (signed) integer to be drawn from the
            distribution (see above for behavior if ``high=None``).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : int or ndarray of ints
            `size`-shaped array of random integers from the appropriate
            distribution, or a single such random int if `size` not provided.

        See Also
        --------
        random.randint : Similar to `random_integers`, only for the half-open
            interval [`low`, `high`), and 0 is the lowest value if `high` is
            omitted.

        Notes
        -----
        To sample from N evenly spaced floating-point numbers between a and b,
        use::

          a + (b - a) * (mkl_random.random_integers(N) - 1) / (N - 1.)

        Examples
        --------
        >>> mkl_random.random_integers(5)
        4
        >>> type(mkl_random.random_integers(5))
        <type 'int'>
        >>> mkl_random.random_integers(5, size=(3.,2.))
        array([[5, 4],
               [3, 3],
               [4, 5]])

        Choose five random numbers from the set of five evenly-spaced
        numbers between 0 and 2.5, inclusive (*i.e.*, from the set
        :math:`{0, 5/8, 10/8, 15/8, 20/8}`):

        >>> 2.5 * (mkl_random.random_integers(5, size=(5,)) - 1) / 4.
        array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ])

        Roll two six sided dice 1000 times and sum the results:

        >>> d1 = mkl_random.random_integers(1, 6, 1000)
        >>> d2 = mkl_random.random_integers(1, 6, 1000)
        >>> dsums = d1 + d2

        Display results as a histogram:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(dsums, 11, normed=True)
        >>> plt.show()

        state_lenexponentialMCG59ononcstandard_gammaohigh
        rayleigh(scale=1.0, size=None)

        Draw samples from a Rayleigh distribution.

        The :math:`\chi` and Weibull distributions are generalizations of the
        Rayleigh.

        Parameters
        ----------
        scale : scalar
            Scale, also equals the mode. Should be >= 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Notes
        -----
        The probability density function for the Rayleigh distribution is

        .. math:: P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}

        The Rayleigh distribution would arise, for example, if the East
        and North components of the wind velocity had identical zero-mean
        Gaussian distributions.  Then the wind speed would have a Rayleigh
        distribution.

        References
        ----------
        .. [1] Brighton Webs Ltd., "Rayleigh Distribution,"
               http://www.brighton-webs.co.uk/distributions/rayleigh.asp
        .. [2] Wikipedia, "Rayleigh distribution"
               http://en.wikipedia.org/wiki/Rayleigh_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram

        >>> values = hist(mkl_random.rayleigh(3, 100000), bins=200, normed=True)

        Wave heights tend to follow a Rayleigh distribution. If the mean wave
        height is 1 meter, what fraction of waves are likely to be larger than 3
        meters?

        >>> meanvalue = 1
        >>> modevalue = np.sqrt(2 / np.pi) * meanvalue
        >>> s = mkl_random.rayleigh(modevalue, 1000000)

        The percentage of waves larger than 3 meters is:

        >>> 100.*sum(s>3)/1000000.
        0.087300000000000003

        RandomState.random_sample (line 1331)The algorithm name does not match content of the bufferrandint_untypedint32
        triangular(left, mode, right, size=None)

        Draw samples from the triangular distribution.

        The triangular distribution is a continuous probability
        distribution with lower limit left, peak at mode, and upper
        limit right. Unlike the other distributions, these parameters
        directly define the shape of the pdf.

        Parameters
        ----------
        left : scalar
            Lower limit.
        mode : scalar
            The value where the peak of the distribution occurs.
            The value should fulfill the condition ``left <= mode <= right``.
        right : scalar
            Upper limit, should be larger than `left`.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The returned samples all lie in the interval [left, right].

        Notes
        -----
        The probability density function for the triangular distribution is

        .. math:: P(x;l, m, r) = \begin{cases}
                  \frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\
                  \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\
                  0& \text{otherwise}.
                  \end{cases}

        The triangular distribution is often used in ill-defined
        problems where the underlying distribution is not known, but
        some knowledge of the limits and mode exists. Often it is used
        in simulations.

        References
        ----------
        .. [1] Wikipedia, "Triangular distribution"
               http://en.wikipedia.org/wiki/Triangular_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram:

        >>> import matplotlib.pyplot as plt
        >>> h = plt.hist(mkl_random.triangular(-3, 0, 8, 100000), bins=200,
        ...              normed=True)
        >>> plt.show()

        basic number generator algorithm must be one of ['replaceomeanlam value too large
        multivariate_normal(mean, cov[, size])

        Draw random samples from a multivariate normal distribution.

        The multivariate normal, multinormal or Gaussian distribution is a
        generalization of the one-dimensional normal distribution to higher
        dimensions.  Such a distribution is specified by its mean and
        covariance matrix.  These parameters are analogous to the mean
        (average or "center") and variance (standard deviation, or "width,"
        squared) of the one-dimensional normal distribution.

        Parameters
        ----------
        mean : 1-D array_like, of length N
            Mean of the N-dimensional distribution.
        cov : 2-D array_like, of shape (N, N)
            Covariance matrix of the distribution. It must be symmetric and
            positive-semidefinite for proper sampling.
        size : int or tuple of ints, optional
            Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
            generated, and packed in an `m`-by-`n`-by-`k` arrangement.  Because
            each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
            If no shape is specified, a single (`N`-D) sample is returned.

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        Notes
        -----
        The mean is a coordinate in N-dimensional space, which represents the
        location where samples are most likely to be generated.  This is
        analogous to the peak of the bell curve for the one-dimensional or
        univariate normal distribution.

        Covariance indicates the level to which two variables vary together.
        From the multivariate normal distribution, we draw N-dimensional
        samples, :math:`X = [x_1, x_2, ... x_N]`.  The covariance matrix
        element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
        The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
        "spread").

        Instead of specifying the full covariance matrix, popular
        approximations include:

          - Spherical covariance (*cov* is a multiple of the identity matrix)
          - Diagonal covariance (*cov* has non-negative elements, and only on
            the diagonal)

        This geometrical property can be seen in two dimensions by plotting
        generated data-points:

        >>> mean = [0, 0]
        >>> cov = [[1, 0], [0, 100]]  # diagonal covariance

        Diagonal covariance means that points are oriented along x or y-axis:

        >>> import matplotlib.pyplot as plt
        >>> x, y = mkl_random.multivariate_normal(mean, cov, 5000).T
        >>> plt.plot(x, y, 'x')
        >>> plt.axis('equal')
        >>> plt.show()

        Note that the covariance matrix must be positive semidefinite (a.k.a.
        nonnegative-definite). Otherwise, the behavior of this method is
        undefined and backwards compatibility is not guaranteed.

        References
        ----------
        .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
               Processes," 3rd ed., New York: McGraw-Hill, 1991.
        .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
               Classification," 2nd ed., New York: Wiley, 2001.

        Examples
        --------
        >>> mean = (1, 2)
        >>> cov = [[1, 0], [0, 1]]
        >>> x = mkl_random.multivariate_normal(mean, cov, (3, 3))
        >>> x.shape
        (3, 3, 2)

        The following is probably true, given that 0.6 is roughly twice the
        standard deviation:

        >>> list((x[0,0,:] - mean) < 0.6)
        [True, True]

        RandomState.randint_untypedstate is expected to be bytes
        logseries(p, size=None)

        Draw samples from a logarithmic series distribution.

        Samples are drawn from a log series distribution with specified
        shape parameter, 0 < ``p`` < 1.

        Parameters
        ----------
        loc : float

        scale : float > 0.

        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
                  where the values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.logser : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Log Series distribution is

        .. math:: P(k) = \frac{-p^k}{k \ln(1-p)},

        where p = probability.

        The log series distribution is frequently used to represent species
        richness and occurrence, first proposed by Fisher, Corbet, and
        Williams in 1943 [2].  It may also be used to model the numbers of
        occupants seen in cars [3].

        References
        ----------
        .. [1] Buzas, Martin A.; Culver, Stephen J.,  Understanding regional
               species diversity through the log series distribution of
               occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
               Volume 5, Number 5, September 1999 , pp. 187-195(9).
        .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
               relation between the number of species and the number of
               individuals in a random sample of an animal population.
               Journal of Animal Ecology, 12:42-58.
        .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
               Data Sets, CRC Press, 1994.
        .. [4] Wikipedia, "Logarithmic-distribution",
               http://en.wikipedia.org/wiki/Logarithmic-distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> a = .6
        >>> s = mkl_random.logseries(a, 10000)
        >>> count, bins, ignored = plt.hist(s)

        #   plot against distribution

        >>> def logseries(k, p):
        ...     return -p**k/(k*log(1-p))
        >>> plt.plot(bins, logseries(bins, a)*count.max()/
                     logseries(bins, a).max(), 'r')
        >>> plt.show()

        p must be 1-dimensionalRandomState.gamma (line 2761)namebool_SFMT19937fmean
        hypergeometric(ngood, nbad, nsample, size=None)

        Draw samples from a Hypergeometric distribution.

        Samples are drawn from a hypergeometric distribution with specified
        parameters, ngood (ways to make a good selection), nbad (ways to make
        a bad selection), and nsample = number of items sampled, which is less
        than or equal to the sum ngood + nbad.

        Parameters
        ----------
        ngood : int or array_like
            Number of ways to make a good selection.  Must be nonnegative.
        nbad : int or array_like
            Number of ways to make a bad selection.  Must be nonnegative.
        nsample : int or array_like
            Number of items sampled.  Must be at least 1 and at most
            ``ngood + nbad``.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(d1, d2, d3)``, then
            ``d1 * d2 * d3`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.hypergeom : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Hypergeometric distribution is

        .. math:: P(x) = \frac{\binom{m}{x}\binom{N-m}{n-x}}{\binom{N}{n}},

        where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n`

        for P(x) the probability of x successes, m = ngood, N = ngood + nbad, and
        n = number of samples.

        Consider an urn with black and white marbles in it, ngood of them
        black and nbad are white. If you draw nsample balls without
        replacement, then the hypergeometric distribution describes the
        distribution of black balls in the drawn sample.

        Note that this distribution is very similar to the binomial
        distribution, except that in this case, samples are drawn without
        replacement, whereas in the Binomial case samples are drawn with
        replacement (or the sample space is infinite). As the sample space
        becomes large, this distribution approaches the binomial.

        References
        ----------
        .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
               and Quigley, 1972.
        .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
               MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/HypergeometricDistribution.html
        .. [3] Wikipedia, "Hypergeometric-distribution",
               http://en.wikipedia.org/wiki/Hypergeometric_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> ngood, nbad, nsamp = 100, 2, 10
        # number of good, number of bad, and number of samples
        >>> s = mkl_random.hypergeometric(ngood, nbad, nsamp, 1000)
        >>> hist(s)
        #   note that it is very unlikely to grab both bad items

        Suppose you have an urn with 15 white and 15 black marbles.
        If you pull 15 marbles at random, how likely is it that
        12 or more of them are one color?

        >>> s = mkl_random.hypergeometric(15, 15, 15, 100000)
        >>> sum(s>=12)/100000. + sum(s<=3)/100000.
        #   answer = 0.003 ... pretty unlikely!

        
        power(a, size=None)

        Draws samples in [0, 1] from a power distribution with positive
        exponent a - 1.

        Also known as the power function distribution.

        Parameters
        ----------
        a : float
            parameter, > 0
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The returned samples lie in [0, 1].

        Raises
        ------
        ValueError
            If a < 1.

        Notes
        -----
        The probability density function is

        .. math:: P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.

        The power function distribution is just the inverse of the Pareto
        distribution. It may also be seen as a special case of the Beta
        distribution.

        It is used, for example, in modeling the over-reporting of insurance
        claims.

        References
        ----------
        .. [1] Christian Kleiber, Samuel Kotz, "Statistical size distributions
               in economics and actuarial sciences", Wiley, 2003.
        .. [2] Heckert, N. A. and Filliben, James J. "NIST Handbook 148:
               Dataplot Reference Manual, Volume 2: Let Subcommands and Library
               Functions", National Institute of Standards and Technology
               Handbook Series, June 2003.
               http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

        Examples
        --------
        Draw samples from the distribution:

        >>> a = 5. # shape
        >>> samples = 1000
        >>> s = mkl_random.power(a, samples)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, bins=30)
        >>> x = np.linspace(0, 1, 100)
        >>> y = a*x**(a-1.)
        >>> normed_y = samples*np.diff(bins)[0]*y
        >>> plt.plot(x, normed_y)
        >>> plt.show()

        Compare the power function distribution to the inverse of the Pareto.

        >>> from scipy import stats
        >>> rvs = mkl_random.power(5, 1000000)
        >>> rvsp = mkl_random.pareto(5, 1000000)
        >>> xx = np.linspace(0,1,100)
        >>> powpdf = stats.powerlaw.pdf(xx,5)

        >>> plt.figure()
        >>> plt.hist(rvs, bins=50, normed=True)
        >>> plt.plot(xx,powpdf,'r-')
        >>> plt.title('mkl_random.power(5)')

        >>> plt.figure()
        >>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
        >>> plt.plot(xx,powpdf,'r-')
        >>> plt.title('inverse of 1 + mkl_random.pareto(5)')

        >>> plt.figure()
        >>> plt.hist(1./(1.+rvsp), bins=50, normed=True)
        >>> plt.plot(xx,powpdf,'r-')
        >>> plt.title('inverse of stats.pareto(5)')

        
        geometric(p, size=None)

        Draw samples from the geometric distribution.

        Bernoulli trials are experiments with one of two outcomes:
        success or failure (an example of such an experiment is flipping
        a coin).  The geometric distribution models the number of trials
        that must be run in order to achieve success.  It is therefore
        supported on the positive integers, ``k = 1, 2, ...``.

        The probability mass function of the geometric distribution is

        .. math:: f(k) = (1 - p)^{k - 1} p

        where `p` is the probability of success of an individual trial.

        Parameters
        ----------
        p : float
            The probability of success of an individual trial.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Samples from the geometric distribution, shaped according to
            `size`.

        Examples
        --------
        Draw ten thousand values from the geometric distribution,
        with the probability of an individual success equal to 0.35:

        >>> z = mkl_random.geometric(p=0.35, size=10000)

        How many trials succeeded after a single run?

        >>> (z == 1).sum() / 10000.
        0.34889999999999999 #random

        buf_ptrRandomState.shuffle (line 5770)RandomState.uniformchisquarelognormalRandomState.poissonintc
        noncentral_f(dfnum, dfden, nonc, size=None)

        Draw samples from the noncentral F distribution.

        Samples are drawn from an F distribution with specified parameters,
        `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
        freedom in denominator), where both parameters > 1.
        `nonc` is the non-centrality parameter.

        Parameters
        ----------
        dfnum : int
            Parameter, should be > 1.
        dfden : int
            Parameter, should be > 1.
        nonc : float
            Parameter, should be >= 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : scalar or ndarray
            Drawn samples.

        Notes
        -----
        When calculating the power of an experiment (power = probability of
        rejecting the null hypothesis when a specific alternative is true) the
        non-central F statistic becomes important.  When the null hypothesis is
        true, the F statistic follows a central F distribution. When the null
        hypothesis is not true, then it follows a non-central F statistic.

        References
        ----------
        .. [1] Weisstein, Eric W. "Noncentral F-Distribution."
               From MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/NoncentralF-Distribution.html
        .. [2] Wikipedia, "Noncentral F distribution",
               http://en.wikipedia.org/wiki/Noncentral_F-distribution

        Examples
        --------
        In a study, testing for a specific alternative to the null hypothesis
        requires use of the Noncentral F distribution. We need to calculate the
        area in the tail of the distribution that exceeds the value of the F
        distribution for the null hypothesis.  We'll plot the two probability
        distributions for comparison.

        >>> dfnum = 3 # between group deg of freedom
        >>> dfden = 20 # within groups degrees of freedom
        >>> nonc = 3.0
        >>> nc_vals = mkl_random.noncentral_f(dfnum, dfden, nonc, 1000000)
        >>> NF = np.histogram(nc_vals, bins=50, normed=True)
        >>> c_vals = mkl_random.f(dfnum, dfden, 1000000)
        >>> F = np.histogram(c_vals, bins=50, normed=True)
        >>> plt.plot(F[1][1:], F[0])
        >>> plt.plot(NF[1][1:], NF[0])
        >>> plt.show()

        uniform_samplesa must be 1-dimensional or an integerRandomState.standard_t (line 3300)
        laplace(loc=0.0, scale=1.0, size=None)

        Draw samples from the Laplace or double exponential distribution with
        specified location (or mean) and scale (decay).

        The Laplace distribution is similar to the Gaussian/normal distribution,
        but is sharper at the peak and has fatter tails. It represents the
        difference between two independent, identically distributed exponential
        random variables.

        Parameters
        ----------
        loc : float, optional
            The position, :math:`\mu`, of the distribution peak.
        scale : float, optional
            :math:`\lambda`, the exponential decay.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or float

        Notes
        -----
        It has the probability density function

        .. math:: f(x; \mu, \lambda) = \frac{1}{2\lambda}
                                       \exp\left(-\frac{|x - \mu|}{\lambda}\right).

        The first law of Laplace, from 1774, states that the frequency
        of an error can be expressed as an exponential function of the
        absolute magnitude of the error, which leads to the Laplace
        distribution. For many problems in economics and health
        sciences, this distribution seems to model the data better
        than the standard Gaussian distribution.

        References
        ----------
        .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
               Mathematical Functions with Formulas, Graphs, and Mathematical
               Tables, 9th printing," New York: Dover, 1972.
        .. [2] Kotz, Samuel, et. al. "The Laplace Distribution and
               Generalizations, " Birkhauser, 2001.
        .. [3] Weisstein, Eric W. "Laplace Distribution."
               From MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/LaplaceDistribution.html
        .. [4] Wikipedia, "Laplace Distribution",
               http://en.wikipedia.org/wiki/Laplace_distribution

        Examples
        --------
        Draw samples from the distribution

        >>> loc, scale = 0., 1.
        >>> s = mkl_random.laplace(loc, scale, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 30, normed=True)
        >>> x = np.arange(-8., 8., .01)
        >>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
        >>> plt.plot(x, pdf)

        Plot Gaussian for comparison:

        >>> g = (1/(scale * np.sqrt(2 * np.pi)) *
        ...      np.exp(-(x - loc)**2 / (2 * scale**2)))
        >>> plt.plot(x,g)

        RandomState.negative_binomial (line 4655)keysdiric
        randint_untyped(low, high=None, size=None, dtype=int)

        Return random integers from `low` (inclusive) to `high` (exclusive).

        Return random integers from the "discrete uniform" distribution of
        the specified dtype in the "half-open" interval [`low`, `high`). If
        `high` is None (the default), then results are from [0, `low`).

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution (unless
            ``high=None``, in which case this parameter is the *highest* such
            integer).
        high : int, optional
            If provided, one above the largest (signed) integer to be drawn
            from the distribution (see above for behavior if ``high=None``).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : int or ndarray of ints
            `size`-shaped array of random integers from the appropriate
            distribution, or a single such random int if `size` not provided.

        See Also
        --------
        random.random_integers : similar to `randint`, only for the closed
            interval [`low`, `high`], and 1 is the lowest value if `high` is
            omitted. In particular, this other one is the one to use to generate
            uniformly distributed discrete non-integers.

        Examples
        --------
        >>> mkl_random.randint(2, size=10)
        array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
        >>> mkl_random.randint(1, size=10)
        array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

        Generate a 2 x 4 array of ints between 0 and 4, inclusive:

        >>> mkl_random.randint(5, size=(2, 4))
        array([[4, 0, 2, 1],
               [3, 2, 2, 0]])

        algo_namecount_nonzeroRandomState.f (line 2855)MCG31bool
        standard_t(df, size=None)

        Draw samples from a standard Student's t distribution with `df` degrees
        of freedom.

        A special case of the hyperbolic distribution.  As `df` gets
        large, the result resembles that of the standard normal
        distribution (`standard_normal`).

        Parameters
        ----------
        df : int
            Degrees of freedom, should be > 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            Drawn samples.

        Notes
        -----
        The probability density function for the t distribution is

        .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
                  \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}

        The t test is based on an assumption that the data come from a
        Normal distribution. The t test provides a way to test whether
        the sample mean (that is the mean calculated from the data) is
        a good estimate of the true mean.

        The derivation of the t-distribution was first published in
        1908 by William Gisset while working for the Guinness Brewery
        in Dublin. Due to proprietary issues, he had to publish under
        a pseudonym, and so he used the name Student.

        References
        ----------
        .. [1] Dalgaard, Peter, "Introductory Statistics With R",
               Springer, 2002.
        .. [2] Wikipedia, "Student's t-distribution"
               http://en.wikipedia.org/wiki/Student's_t-distribution

        Examples
        --------
        From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
        women in Kj is:

        >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
        ...                    7515, 8230, 8770])

        Does their energy intake deviate systematically from the recommended
        value of 7725 kJ?

        We have 10 degrees of freedom, so is the sample mean within 95% of the
        recommended value?

        >>> s = mkl_random.standard_t(10, size=100000)
        >>> np.mean(intake)
        6753.636363636364
        >>> intake.std(ddof=1)
        1142.1232221373727

        Calculate the t statistic, setting the ddof parameter to the unbiased
        value so the divisor in the standard deviation will be degrees of
        freedom, N-1.

        >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
        >>> import matplotlib.pyplot as plt
        >>> h = plt.hist(s, bins=100, normed=True)

        For a one-sided t-test, how far out in the distribution does the t
        statistic appear?

        >>> np.sum(s<t) / float(len(s))
        0.0090699999999999999  #random

        So the p-value is about 0.009, which says the null hypothesis has a
        probability of about 99% of being true.

        The legacy state input algorithm must be 'MT19937'RandomState.zipfalpha_arrRandomState._rand_uint8RandomState.lognormalint8olamRandomState._rand_int8hypergeometricscale__pyx_vtable__seed
        permutation(x)

        Randomly permute a sequence, or return a permuted range.

        If `x` is a multi-dimensional array, it is only shuffled along its
        first index.

        Parameters
        ----------
        x : int or array_like
            If `x` is an integer, randomly permute ``np.arange(x)``.
            If `x` is an array, make a copy and shuffle the elements
            randomly.

        Returns
        -------
        out : ndarray
            Permuted sequence or array range.

        Examples
        --------
        >>> mkl_random.permutation(10)
        array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6])

        >>> mkl_random.permutation([1, 4, 9, 12, 15])
        array([15,  1,  9,  4, 12])

        >>> arr = np.arange(9).reshape((3, 3))
        >>> mkl_random.permutation(arr)
        array([[6, 7, 8],
               [0, 1, 2],
               [3, 4, 5]])

        transpose
        zipf(a, size=None)

        Draw samples from a Zipf distribution.

        Samples are drawn from a Zipf distribution with specified parameter
        `a` > 1.

        The Zipf distribution (also known as the zeta distribution) is a
        continuous probability distribution that satisfies Zipf's law: the
        frequency of an item is inversely proportional to its rank in a
        frequency table.

        Parameters
        ----------
        a : float > 1
            Distribution parameter.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : scalar or ndarray
            The returned samples are greater than or equal to one.

        See Also
        --------
        scipy.stats.distributions.zipf : probability density function,
            distribution, or cumulative density function, etc.

        Notes
        -----
        The probability density for the Zipf distribution is

        .. math:: p(x) = \frac{x^{-a}}{\zeta(a)},

        where :math:`\zeta` is the Riemann Zeta function.

        It is named for the American linguist George Kingsley Zipf, who noted
        that the frequency of any word in a sample of a language is inversely
        proportional to its rank in the frequency table.

        References
        ----------
        .. [1] Zipf, G. K., "Selected Studies of the Principle of Relative
               Frequency in Language," Cambridge, MA: Harvard Univ. Press,
               1932.

        Examples
        --------
        Draw samples from the distribution:

        >>> a = 2. # parameter
        >>> s = mkl_random.zipf(a, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> import scipy.special as sps
        Truncate s values at 50 so plot is interesting
        >>> count, bins, ignored = plt.hist(s[s<50], 50, normed=True)
        >>> x = np.arange(1., 50.)
        >>> y = x**(-a)/sps.zetac(a)
        >>> plt.plot(x, y/max(y), linewidth=2, color='r')
        >>> plt.show()

        RandomState.beta
        rand(d0, d1, ..., dn)

        Random values in a given shape.

        Create an array of the given shape and propagate it with
        random samples from a uniform distribution
        over ``[0, 1)``.

        Parameters
        ----------
        d0, d1, ..., dn : int, optional
            The dimensions of the returned array, should all be positive.
            If no argument is given a single Python float is returned.

        Returns
        -------
        out : ndarray, shape ``(d0, d1, ..., dn)``
            Random values.

        See Also
        --------
        random

        Notes
        -----
        This is a convenience function. If you want an interface that
        takes a shape-tuple as the first argument, refer to
        mkl_random.random_sample .

        Examples
        --------
        >>> mkl_random.rand(3,2)
        array([[ 0.14022471,  0.96360618],  #random
               [ 0.37601032,  0.25528411],  #random
               [ 0.49313049,  0.94909878]]) #random

        lntot
        random_sample(size=None)

        Return random floats in the half-open interval [0.0, 1.0).

        Results are from the "continuous uniform" distribution over the
        stated interval.  To sample :math:`Unif[a, b), b > a` multiply
        the output of `random_sample` by `(b-a)` and add `a`::

          (b - a) * random_sample() + a

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : float or ndarray of floats
            Array of random floats of shape `size` (unless ``size=None``, in which
            case a single float is returned).

        Examples
        --------
        >>> mkl_random.random_sample()
        0.47108547995356098
        >>> type(mkl_random.random_sample())
        <type 'float'>
        >>> mkl_random.random_sample((5,))
        array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428])

        Three-by-two array of random numbers from [-5, 0):

        >>> 5 * mkl_random.random_sample((3, 2)) - 5
        array([[-3.99149989, -0.52338984],
               [-2.99091858, -0.79479508],
               [-1.23204345, -1.75224494]])

        reversedpop_sizeR250size_parse_brng_argumentch must be a square lower triangular 2-dimensional array or a row-packed one-dimensional representation of suchRandomState.noncentral_f (line 2961)datamean <= 0.0nstreams
        standard_exponential(size=None)

        Draw samples from the standard exponential distribution.

        `standard_exponential` is identical to the exponential distribution
        with a scale parameter of 1.

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : float or ndarray
            Drawn samples.

        Examples
        --------
        Output a 3x8000 array:

        >>> n = mkl_random.standard_exponential((3, 8000))

        RandomState.hypergeometricn > 2147483647index
        _rand_int16(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        __test__
        _rand_bool(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        a and p must have same size
        _rand_int8(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        nbadfinfoitemsizeflowRandomState.leapfrog
        standard_cauchy(size=None)

        Draw samples from a standard Cauchy distribution with mode = 0.

        Also known as the Lorentz distribution.

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The drawn samples.

        Notes
        -----
        The probability density function for the full Cauchy distribution is

        .. math:: P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
                  (\frac{x-x_0}{\gamma})^2 \bigr] }

        and the Standard Cauchy distribution just sets :math:`x_0=0` and
        :math:`\gamma=1`

        The Cauchy distribution arises in the solution to the driven harmonic
        oscillator problem, and also describes spectral line broadening. It
        also describes the distribution of values at which a line tilted at
        a random angle will cut the x axis.

        When studying hypothesis tests that assume normality, seeing how the
        tests perform on data from a Cauchy distribution is a good indicator of
        their sensitivity to a heavy-tailed distribution, since the Cauchy looks
        very much like a Gaussian distribution, but with heavier tails.

        References
        ----------
        .. [1] NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy
              Distribution",
              http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
        .. [2] Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A
              Wolfram Web Resource.
              http://mathworld.wolfram.com/CauchyDistribution.html
        .. [3] Wikipedia, "Cauchy distribution"
              http://en.wikipedia.org/wiki/Cauchy_distribution

        Examples
        --------
        Draw samples and plot the distribution:

        >>> s = mkl_random.standard_cauchy(1000000)
        >>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
        >>> plt.hist(s, bins=100)
        >>> plt.show()

        random_sampleisenabled_choose_randint_typecopy
        get_state()

        Return a tuple representing the internal state of the generator.

        For more details, see `set_state`.

        Returns
        -------
        out : tuple(str, bytes)
            The returned tuple has the following items:

            1. the string, defaulting to 'MT19937', specifying the
               basic psedo-random number generation algorithm.
            2. a bytes object holding content of Intel MKL's stream for the
               given BRNG.

        See Also
        --------
        set_state

        Notes
        -----
        `set_state` and `get_state` are not needed to work with any of the
        random distributions in NumPy. If the internal state is manually altered,
        the user should know exactly what he/she is doing.

        References
        -----
        MKL Documentation: https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl.html

        
        multinomial(n, pvals, size=None, method='ICDF')

        Draw samples from a multinomial distribution.

        The multinomial distribution is a multivariate generalisation of the
        binomial distribution.  Take an experiment with one of ``p``
        possible outcomes.  An example of such an experiment is throwing a dice,
        where the outcome can be 1 through 6.  Each sample drawn from the
        distribution represents `n` such experiments.  Its values,
        ``X_i = [X_0, X_1, ..., X_p]``, represent the number of times the
        outcome was ``i``.

        Parameters
        ----------
        n : int
            Number of experiments.
        pvals : sequence of floats, length p
            Probabilities of each of the ``p`` different outcomes.  These
            should sum to 1 (however, the last element is always assumed to
            account for the remaining probability, as long as
            ``sum(pvals[:-1]) <= 1)``.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        Examples
        --------
        Throw a dice 20 times:

        >>> mkl_random.multinomial(20, [1/6.]*6, size=1)
        array([[4, 1, 7, 5, 2, 1]])

        It landed 4 times on 1, once on 2, etc.

        Now, throw the dice 20 times, and 20 times again:

        >>> mkl_random.multinomial(20, [1/6.]*6, size=2)
        array([[3, 4, 3, 3, 4, 3],
               [2, 4, 3, 4, 0, 7]])

        For the first run, we threw 3 times 1, 4 times 2, etc.  For the second,
        we threw 2 times 1, 4 times 2, etc.

        A loaded die is more likely to land on number 6:

        >>> mkl_random.multinomial(100, [1/7.]*5 + [2./7.])
        array([11, 16, 14, 17, 16, 26])

        The probability inputs should be normalized. As an implementation
        detail, the value of the last entry is ignored and assumed to take
        up any leftover probability mass, but this should not be relied on.
        A biased coin which has twice as much weight on one side as on the
        other should be sampled like so:

        >>> mkl_random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
        array([38, 62])

        not like:

        >>> mkl_random.multinomial(100, [1.0, 2.0])  # WRONG
        array([100,   0])

        onsampletotsize
        uniform(low=0.0, high=1.0, size=None)

        Draw samples from a uniform distribution.

        Samples are uniformly distributed over the half-open interval
        ``[low, high)`` (includes low, but excludes high).  In other words,
        any value within the given interval is equally likely to be drawn
        by `uniform`.

        Parameters
        ----------
        low : float, optional
            Lower boundary of the output interval.  All values generated will be
            greater than or equal to low.  The default value is 0.
        high : float
            Upper boundary of the output interval.  All values generated will be
            less than high.  The default value is 1.0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Drawn samples, with shape `size`.

        See Also
        --------
        randint : Discrete uniform distribution, yielding integers.
        random_integers : Discrete uniform distribution over the closed
                          interval ``[low, high]``.
        random_sample : Floats uniformly distributed over ``[0, 1)``.
        random : Alias for `random_sample`.
        rand : Convenience function that accepts dimensions as input, e.g.,
               ``rand(2,2)`` would generate a 2-by-2 array of floats,
               uniformly distributed over ``[0, 1)``.

        Notes
        -----
        The probability density function of the uniform distribution is

        .. math:: p(x) = \frac{1}{b - a}

        anywhere within the interval ``[a, b)``, and zero elsewhere.

        Examples
        --------
        Draw samples from the distribution:

        >>> s = mkl_random.uniform(-1,0,1000)

        All values are within the given interval:

        >>> np.all(s >= -1)
        True
        >>> np.all(s < 0)
        True

        Display the histogram of the samples, along with the
        probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 15, normed=True)
        >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
        >>> plt.show()

        iinfo__getstate__RandomState.power (line 3716)greater
        _rand_int64(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        operatormean and cov must have same lengthRandomState._rand_booldirichletlaplaceerrcodeRandomState.multinomialparr
        logseries(p, size=None)

        Draw samples from a logarithmic series distribution.

        Samples are drawn from a log series distribution with specified
        shape parameter, 0 < ``p`` < 1.

        Parameters
        ----------
        loc : float

        scale : float > 0.

        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
                  where the values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.logser : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Log Series distribution is

        .. math:: P(k) = \frac{-p^k}{k \ln(1-p)},

        where p = probability.

        The log series distribution is frequently used to represent species
        richness and occurrence, first proposed by Fisher, Corbet, and
        Williams in 1943 [2].  It may also be used to model the numbers of
        occupants seen in cars [3].

        References
        ----------
        .. [1] Buzas, Martin A.; Culver, Stephen J.,  Understanding regional
               species diversity through the log series distribution of
               occurrences: BIODIVERSITY RESEARCH Diversity & Distributions,
               Volume 5, Number 5, September 1999 , pp. 187-195(9).
        .. [2] Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The
               relation between the number of species and the number of
               individuals in a random sample of an animal population.
               Journal of Animal Ecology, 12:42-58.
        .. [3] D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small
               Data Sets, CRC Press, 1994.
        .. [4] Wikipedia, "Logarithmic-distribution",
               http://en.wikipedia.org/wiki/Logarithmic-distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> a = .6
        >>> s = mkl_random.logseries(a, 10000)
        >>> count, bins, ignored = plt.hist(s)

        #   plot against distribution

        >>> def logseries(k, p):
        ...     return -p**k/(k*log(1-p))
        >>> plt.plot(bins, logseries(bins, a)*count.max()/
                     logseries(bins, a).max(), 'r')
        >>> plt.show()

        RandomState.standard_titem
        randint(low, high=None, size=None, dtype=int)

        Return random integers from `low` (inclusive) to `high` (exclusive).

        Return random integers from the "discrete uniform" distribution of
        the specified dtype in the "half-open" interval [`low`, `high`). If
        `high` is None (the default), then results are from [0, `low`).

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution (unless
            ``high=None``, in which case this parameter is the *highest* such
            integer).
        high : int, optional
            If provided, one above the largest (signed) integer to be drawn
            from the distribution (see above for behavior if ``high=None``).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        dtype : dtype, optional
            Desired dtype of the result. All dtypes are determined by their
            name, i.e., 'int64', 'int', etc, so byteorder is not available
            and a specific precision may have different C types depending
            on the platform. The default value is 'np.int'.

            .. versionadded:: 1.11.0

        Returns
        -------
        out : int or ndarray of ints
            `size`-shaped array of random integers from the appropriate
            distribution, or a single such random int if `size` not provided.

        See Also
        --------
        random.random_integers : similar to `randint`, only for the closed
            interval [`low`, `high`], and 1 is the lowest value if `high` is
            omitted. In particular, this other one is the one to use to generate
            uniformly distributed discrete non-integers.

        Examples
        --------
        >>> mkl_random.randint(2, size=10)
        array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
        >>> mkl_random.randint(1, size=10)
        array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

        Generate a 2 x 4 array of ints between 0 and 4, inclusive:

        >>> mkl_random.randint(5, size=(2, 4))
        array([[4, 0, 2, 1],
               [3, 2, 2, 0]])

        RandomState.zipf (line 4847)rand_rand_int8
        noncentral_chisquare(df, nonc, size=None)

        Draw samples from a noncentral chi-square distribution.

        The noncentral :math:`\chi^2` distribution is a generalisation of
        the :math:`\chi^2` distribution.

        Parameters
        ----------
        df : int
            Degrees of freedom, should be > 0 as of Numpy 1.10,
            should be > 1 for earlier versions.
        nonc : float
            Non-centrality, should be non-negative.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Notes
        -----
        The probability density function for the noncentral Chi-square
        distribution is

        .. math:: P(x;df,nonc) = \sum^{\infty}_{i=0}
                               \frac{e^{-nonc/2}(nonc/2)^{i}}{i!}
                               \P_{Y_{df+2i}}(x),

        where :math:`Y_{q}` is the Chi-square with q degrees of freedom.

        In Delhi (2007), it is noted that the noncentral chi-square is
        useful in bombing and coverage problems, the probability of
        killing the point target given by the noncentral chi-squared
        distribution.

        References
        ----------
        .. [1] Delhi, M.S. Holla, "On a noncentral chi-square distribution in
               the analysis of weapon systems effectiveness", Metrika,
               Volume 15, Number 1 / December, 1970.
        .. [2] Wikipedia, "Noncentral chi-square distribution"
               http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram

        >>> import matplotlib.pyplot as plt
        >>> values = plt.hist(mkl_random.noncentral_chisquare(3, 20, 100000),
        ...                   bins=200, normed=True)
        >>> plt.show()

        Draw values from a noncentral chisquare with very small noncentrality,
        and compare to a chisquare.

        >>> plt.figure()
        >>> values = plt.hist(mkl_random.noncentral_chisquare(3, .0000001, 100000),
        ...                   bins=np.arange(0., 25, .1), normed=True)
        >>> values2 = plt.hist(mkl_random.chisquare(3, 100000),
        ...                    bins=np.arange(0., 25, .1), normed=True)
        >>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
        >>> plt.show()

        Demonstrate how large values of non-centrality lead to a more symmetric
        distribution.

        >>> plt.figure()
        >>> values = plt.hist(mkl_random.noncentral_chisquare(3, 20, 100000),
        ...                   bins=200, normed=True)
        >>> plt.show()

        RandomState.gamma_rand_int64RandomState.standard_gammaRandomState.bytes (line 1859)
        uniform(low=0.0, high=1.0, size=None)

        Draw samples from a uniform distribution.

        Samples are uniformly distributed over the half-open interval
        ``[low, high)`` (includes low, but excludes high).  In other words,
        any value within the given interval is equally likely to be drawn
        by `uniform`.

        Parameters
        ----------
        low : float, optional
            Lower boundary of the output interval.  All values generated will be
            greater than or equal to low.  The default value is 0.
        high : float
            Upper boundary of the output interval.  All values generated will be
            less than high.  The default value is 1.0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Drawn samples, with shape `size`.

        See Also
        --------
        randint : Discrete uniform distribution, yielding integers.
        random_integers : Discrete uniform distribution over the closed
                          interval ``[low, high]``.
        random_sample : Floats uniformly distributed over ``[0, 1)``.
        random : Alias for `random_sample`.
        rand : Convenience function that accepts dimensions as input, e.g.,
               ``rand(2,2)`` would generate a 2-by-2 array of floats,
               uniformly distributed over ``[0, 1)``.

        Notes
        -----
        The probability density function of the uniform distribution is

        .. math:: p(x) = \frac{1}{b - a}

        anywhere within the interval ``[a, b)``, and zero elsewhere.

        Examples
        --------
        Draw samples from the distribution:

        >>> s = mkl_random.uniform(-1,0,1000)

        All values are within the given interval:

        >>> np.all(s >= -1)
        True
        >>> np.all(s < 0)
        True

        Display the histogram of the samples, along with the
        probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 15, normed=True)
        >>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
        >>> plt.show()

        DeprecationWarningRandomState.seedres_data
        dirichlet(alpha, size=None)

        Draw samples from the Dirichlet distribution.

        Draw `size` samples of dimension k from a Dirichlet distribution. A
        Dirichlet-distributed random variable can be seen as a multivariate
        generalization of a Beta distribution. Dirichlet pdf is the conjugate
        prior of a multinomial in Bayesian inference.

        Parameters
        ----------
        alpha : array
            Parameter of the distribution (k dimension for sample of
            dimension k).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray,
            The drawn samples, of shape (size, alpha.ndim).

        Notes
        -----
        .. math:: X \approx \prod_{i=1}^{k}{x^{\alpha_i-1}_i}

        Uses the following property for computation: for each dimension,
        draw a random sample y_i from a standard gamma generator of shape
        `alpha_i`, then
        :math:`X = \frac{1}{\sum_{i=1}^k{y_i}} (y_1, \ldots, y_n)` is
        Dirichlet distributed.

        References
        ----------
        .. [1] David McKay, "Information Theory, Inference and Learning
               Algorithms," chapter 23,
               http://www.inference.phy.cam.ac.uk/mackay/
        .. [2] Wikipedia, "Dirichlet distribution",
               http://en.wikipedia.org/wiki/Dirichlet_distribution

        Examples
        --------
        Taking an example cited in Wikipedia, this distribution can be used if
        one wanted to cut strings (each of initial length 1.0) into K pieces
        with different lengths, where each piece had, on average, a designated
        average length, but allowing some variation in the relative sizes of
        the pieces.

        >>> s = mkl_random.dirichlet((10, 5, 3), 20).transpose()

        >>> plt.barh(range(20), s[0])
        >>> plt.barh(range(20), s[1], left=s[0], color='g')
        >>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
        >>> plt.title("Lengths of Strings")

        RandomState.skipaheadrandnbytesPtrlogseries
        f(dfnum, dfden, size=None)

        Draw samples from an F distribution.

        Samples are drawn from an F distribution with specified parameters,
        `dfnum` (degrees of freedom in numerator) and `dfden` (degrees of
        freedom in denominator), where both parameters should be greater than
        zero.

        The random variate of the F distribution (also known as the
        Fisher distribution) is a continuous probability distribution
        that arises in ANOVA tests, and is the ratio of two chi-square
        variates.

        Parameters
        ----------
        dfnum : float
            Degrees of freedom in numerator. Should be greater than zero.
        dfden : float
            Degrees of freedom in denominator. Should be greater than zero.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            Samples from the Fisher distribution.

        See Also
        --------
        scipy.stats.distributions.f : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The F statistic is used to compare in-group variances to between-group
        variances. Calculating the distribution depends on the sampling, and
        so it is a function of the respective degrees of freedom in the
        problem.  The variable `dfnum` is the number of samples minus one, the
        between-groups degrees of freedom, while `dfden` is the within-groups
        degrees of freedom, the sum of the number of samples in each group
        minus the number of groups.

        References
        ----------
        .. [1] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
               Fifth Edition, 2002.
        .. [2] Wikipedia, "F-distribution",
               http://en.wikipedia.org/wiki/F-distribution

        Examples
        --------
        An example from Glantz[1], pp 47-40:

        Two groups, children of diabetics (25 people) and children from people
        without diabetes (25 controls). Fasting blood glucose was measured,
        case group had a mean value of 86.1, controls had a mean value of
        82.2. Standard deviations were 2.09 and 2.49 respectively. Are these
        data consistent with the null hypothesis that the parents diabetic
        status does not affect their children's blood glucose levels?
        Calculating the F statistic from the data gives a value of 36.01.

        Draw samples from the distribution:

        >>> dfnum = 1. # between group degrees of freedom
        >>> dfden = 48. # within groups degrees of freedom
        >>> s = mkl_random.f(dfnum, dfden, 1000)

        The lower bound for the top 1% of the samples is :

        >>> sort(s)[-10]
        7.61988120985

        So there is about a 1% chance that the F statistic will exceed 7.62,
        the measured value is 36, so the null hypothesis is rejected at the 1%
        level.

        lam < 0BOXMULLERsearchsortedRandomState.poisson (line 4751)zipfThis function is deprecated. Please call randint(1, {low} + 1) insteadvonmisesRandomState._choose_randint_typegivenmean and ch must have consistent shapes
        geometric(p, size=None)

        Draw samples from the geometric distribution.

        Bernoulli trials are experiments with one of two outcomes:
        success or failure (an example of such an experiment is flipping
        a coin).  The geometric distribution models the number of trials
        that must be run in order to achieve success.  It is therefore
        supported on the positive integers, ``k = 1, 2, ...``.

        The probability mass function of the geometric distribution is

        .. math:: f(k) = (1 - p)^{k - 1} p

        where `p` is the probability of success of an individual trial.

        Parameters
        ----------
        p : float
            The probability of success of an individual trial.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Samples from the geometric distribution, shaped according to
            `size`.

        Examples
        --------
        Draw ten thousand values from the geometric distribution,
        with the probability of an individual success equal to 0.35:

        >>> z = mkl_random.geometric(p=0.35, size=10000)

        How many trials succeeded after a single run?

        >>> (z == 1).sum() / 10000.
        0.34889999999999999 #random

        OverflowError
        standard_exponential(size=None)

        Draw samples from the standard exponential distribution.

        `standard_exponential` is identical to the exponential distribution
        with a scale parameter of 1.

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : float or ndarray
            Drawn samples.

        Examples
        --------
        Output a 3x8000 array:

        >>> n = mkl_random.standard_exponential((3, 8000))

        emptyfnoncmultinomialpoisson_lam_maxolow
        randint_untyped(low, high=None, size=None, dtype=int)

        Return random integers from `low` (inclusive) to `high` (exclusive).

        Return random integers from the "discrete uniform" distribution of
        the specified dtype in the "half-open" interval [`low`, `high`). If
        `high` is None (the default), then results are from [0, `low`).

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution (unless
            ``high=None``, in which case this parameter is the *highest* such
            integer).
        high : int, optional
            If provided, one above the largest (signed) integer to be drawn
            from the distribution (see above for behavior if ``high=None``).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : int or ndarray of ints
            `size`-shaped array of random integers from the appropriate
            distribution, or a single such random int if `size` not provided.

        See Also
        --------
        random.random_integers : similar to `randint`, only for the closed
            interval [`low`, `high`], and 1 is the lowest value if `high` is
            omitted. In particular, this other one is the one to use to generate
            uniformly distributed discrete non-integers.

        Examples
        --------
        >>> mkl_random.randint(2, size=10)
        array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
        >>> mkl_random.randint(1, size=10)
        array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

        Generate a 2 x 4 array of ints between 0 and 4, inclusive:

        >>> mkl_random.randint(5, size=(2, 4))
        array([[4, 0, 2, 1],
               [3, 2, 2, 0]])

        RandomState.noncentral_chisquare (line 3138)foundRandomState.standard_exponential
        logistic(loc=0.0, scale=1.0, size=None)

        Draw samples from a logistic distribution.

        Samples are drawn from a logistic distribution with specified
        parameters, loc (location or mean, also median), and scale (>0).

        Parameters
        ----------
        loc : float

        scale : float > 0.

        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
                  where the values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.logistic : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Logistic distribution is

        .. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},

        where :math:`\mu` = location and :math:`s` = scale.

        The Logistic distribution is used in Extreme Value problems where it
        can act as a mixture of Gumbel distributions, in Epidemiology, and by
        the World Chess Federation (FIDE) where it is used in the Elo ranking
        system, assuming the performance of each player is a logistically
        distributed random variable.

        References
        ----------
        .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of
               Extreme Values, from Insurance, Finance, Hydrology and Other
               Fields," Birkhauser Verlag, Basel, pp 132-133.
        .. [2] Weisstein, Eric W. "Logistic Distribution." From
               MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/LogisticDistribution.html
        .. [3] Wikipedia, "Logistic-distribution",
               http://en.wikipedia.org/wiki/Logistic_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> loc, scale = 10, 1
        >>> s = mkl_random.logistic(loc, scale, 10000)
        >>> count, bins, ignored = plt.hist(s, bins=50)

        #   plot against distribution

        >>> def logist(x, loc, scale):
        ...     return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2)
        >>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\
        ... logist(bins, loc, scale).max())
        >>> plt.show()

        scale <= 0uint8
        wald(mean, scale, size=None)

        Draw samples from a Wald, or inverse Gaussian, distribution.

        As the scale approaches infinity, the distribution becomes more like a
        Gaussian. Some references claim that the Wald is an inverse Gaussian
        with mean equal to 1, but this is by no means universal.

        The inverse Gaussian distribution was first studied in relationship to
        Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian
        because there is an inverse relationship between the time to cover a
        unit distance and distance covered in unit time.

        Parameters
        ----------
        mean : scalar
            Distribution mean, should be > 0.
        scale : scalar
            Scale parameter, should be >= 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            Drawn sample, all greater than zero.

        Notes
        -----
        The probability density function for the Wald distribution is

        .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^
                                    \frac{-scale(x-mean)^2}{2\cdotp mean^2x}

        As noted above the inverse Gaussian distribution first arise
        from attempts to model Brownian motion. It is also a
        competitor to the Weibull for use in reliability modeling and
        modeling stock returns and interest rate processes.

        References
        ----------
        .. [1] Brighton Webs Ltd., Wald Distribution,
               http://www.brighton-webs.co.uk/distributions/wald.asp
        .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian
               Distribution: Theory : Methodology, and Applications", CRC Press,
               1988.
        .. [3] Wikipedia, "Wald distribution"
               http://en.wikipedia.org/wiki/Wald_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram:

        >>> import matplotlib.pyplot as plt
        >>> h = plt.hist(mkl_random.wald(3, 2, 100000), bins=200, normed=True)
        >>> plt.show()

        a must be 1-dimensionalside
        vonmises(mu, kappa, size=None)

        Draw samples from a von Mises distribution.

        Samples are drawn from a von Mises distribution with specified mode
        (mu) and dispersion (kappa), on the interval [-pi, pi].

        The von Mises distribution (also known as the circular normal
        distribution) is a continuous probability distribution on the unit
        circle.  It may be thought of as the circular analogue of the normal
        distribution.

        Parameters
        ----------
        mu : float
            Mode ("center") of the distribution.
        kappa : float
            Dispersion of the distribution, has to be >=0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : scalar or ndarray
            The returned samples, which are in the interval [-pi, pi].

        See Also
        --------
        scipy.stats.distributions.vonmises : probability density function,
            distribution, or cumulative density function, etc.

        Notes
        -----
        The probability density for the von Mises distribution is

        .. math:: p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},

        where :math:`\mu` is the mode and :math:`\kappa` the dispersion,
        and :math:`I_0(\kappa)` is the modified Bessel function of order 0.

        The von Mises is named for Richard Edler von Mises, who was born in
        Austria-Hungary, in what is now the Ukraine.  He fled to the United
        States in 1939 and became a professor at Harvard.  He worked in
        probability theory, aerodynamics, fluid mechanics, and philosophy of
        science.

        References
        ----------
        .. [1] Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of
               Mathematical Functions with Formulas, Graphs, and Mathematical
               Tables, 9th printing," New York: Dover, 1972.
        .. [2] von Mises, R., "Mathematical Theory of Probability
               and Statistics", New York: Academic Press, 1964.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, kappa = 0.0, 4.0 # mean and dispersion
        >>> s = mkl_random.vonmises(mu, kappa, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> from scipy.special import i0
        >>> plt.hist(s, 50, normed=True)
        >>> x = np.linspace(-np.pi, np.pi, num=51)
        >>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))
        >>> plt.plot(x, y, linewidth=2, color='r')
        >>> plt.show()

        tomaxintunique_indices
        seed(seed=None, brng=None)

        Seed the generator.

        This method is called when `RandomState` is initialized. It can be
        called again to re-seed the generator. For details, see `RandomState`.

        Parameters
        ----------
        seed : int or array_like, optional
            Seed for `RandomState`.
            Must be convertible to 32 bit unsigned integers.
        brng : {'MT19937', 'SFMT19937', 'MT2203', 'R250', 'WH', 'MCG31',
                'MCG59', 'MRG32K3A', 'PHILOX4X32X10', 'NONDETERM',
                'ARS5', None}, optional
            basic pseudo-random number generation algorithms, or non-deterministic
            hardware-based generator, provided by Intel MKL. Use `brng==None` to keep
            the `brng` specified during construction of this class instance.

        See Also
        --------
        RandomState

        References
        --------
        MKL Documentation: https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl.html

        TypeErrorbrng_tokenRandomState.randn
        beta(a, b, size=None)

        Draw samples from a Beta distribution.

        The Beta distribution is a special case of the Dirichlet distribution
        and is related to the Gamma distribution.  It has the probability
        distribution function

        .. math:: f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1}
                                                         (1 - x)^{\beta - 1},

        where the normalisation, B, is the beta function,

        .. math:: B(\alpha, \beta) = \int_0^1 t^{\alpha - 1}
                                     (1 - t)^{\beta - 1} dt.

        It is often seen in Bayesian inference and order statistics.

        Parameters
        ----------
        a : float
            Alpha, non-negative.
        b : float
            Beta, non-negative.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Array of the given shape, containing values drawn from a
            Beta distribution.

        p <= 0.0
        wald(mean, scale, size=None)

        Draw samples from a Wald, or inverse Gaussian, distribution.

        As the scale approaches infinity, the distribution becomes more like a
        Gaussian. Some references claim that the Wald is an inverse Gaussian
        with mean equal to 1, but this is by no means universal.

        The inverse Gaussian distribution was first studied in relationship to
        Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian
        because there is an inverse relationship between the time to cover a
        unit distance and distance covered in unit time.

        Parameters
        ----------
        mean : scalar
            Distribution mean, should be > 0.
        scale : scalar
            Scale parameter, should be >= 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            Drawn sample, all greater than zero.

        Notes
        -----
        The probability density function for the Wald distribution is

        .. math:: P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^
                                    \frac{-scale(x-mean)^2}{2\cdotp mean^2x}

        As noted above the inverse Gaussian distribution first arise
        from attempts to model Brownian motion. It is also a
        competitor to the Weibull for use in reliability modeling and
        modeling stock returns and interest rate processes.

        References
        ----------
        .. [1] Brighton Webs Ltd., Wald Distribution,
               http://www.brighton-webs.co.uk/distributions/wald.asp
        .. [2] Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian
               Distribution: Theory : Methodology, and Applications", CRC Press,
               1988.
        .. [3] Wikipedia, "Wald distribution"
               http://en.wikipedia.org/wiki/Wald_distribution

        Examples
        --------
        Draw values from the distribution and plot the histogram:

        >>> import matplotlib.pyplot as plt
        >>> h = plt.hist(mkl_random.wald(3, 2, 100000), bins=200, normed=True)
        >>> plt.show()

        Seed must be between 0 and 4294967295fmodeThe generator index {actual} is not between 0 and {max}, index {choice} will be used.left == rightmean_dataRandomState.triangular
        randint(low, high=None, size=None, dtype=int)

        Return random integers from `low` (inclusive) to `high` (exclusive).

        Return random integers from the "discrete uniform" distribution of
        the specified dtype in the "half-open" interval [`low`, `high`). If
        `high` is None (the default), then results are from [0, `low`).

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution (unless
            ``high=None``, in which case this parameter is the *highest* such
            integer).
        high : int, optional
            If provided, one above the largest (signed) integer to be drawn
            from the distribution (see above for behavior if ``high=None``).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        dtype : dtype, optional
            Desired dtype of the result. All dtypes are determined by their
            name, i.e., 'int64', 'int', etc, so byteorder is not available
            and a specific precision may have different C types depending
            on the platform. The default value is 'np.int'.

            .. versionadded:: 1.11.0

        Returns
        -------
        out : int or ndarray of ints
            `size`-shaped array of random integers from the appropriate
            distribution, or a single such random int if `size` not provided.

        See Also
        --------
        random.random_integers : similar to `randint`, only for the closed
            interval [`low`, `high`], and 1 is the lowest value if `high` is
            omitted. In particular, this other one is the one to use to generate
            uniformly distributed discrete non-integers.

        Examples
        --------
        >>> mkl_random.randint(2, size=10)
        array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])
        >>> mkl_random.randint(1, size=10)
        array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

        Generate a 2 x 4 array of ints between 0 and 4, inclusive:

        >>> mkl_random.randint(5, size=(2, 4))
        array([[4, 0, 2, 1],
               [3, 2, 2, 0]])

        args
        negative_binomial(n, p, size=None)

        Draw samples from a negative binomial distribution.

        Samples are drawn from a negative binomial distribution with specified
        parameters, `n` trials and `p` probability of success where `n` is an
        integer > 0 and `p` is in the interval [0, 1].

        Parameters
        ----------
        n : int
            Parameter, > 0.
        p : float
            Parameter, >= 0 and <=1.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : int or ndarray of ints
            Drawn samples.

        Notes
        -----
        The probability density for the negative binomial distribution is

        .. math:: P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N},

        where :math:`n-1` is the number of successes, :math:`p` is the
        probability of success, and :math:`N+n-1` is the number of trials.
        The negative binomial distribution gives the probability of n-1
        successes and N failures in N+n-1 trials, and success on the (N+n)th
        trial.

        If one throws a die repeatedly until the third time a "1" appears,
        then the probability distribution of the number of non-"1"s that
        appear before the third "1" is a negative binomial distribution.

        References
        ----------
        .. [1] Weisstein, Eric W. "Negative Binomial Distribution." From
               MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/NegativeBinomialDistribution.html
        .. [2] Wikipedia, "Negative binomial distribution",
               http://en.wikipedia.org/wiki/Negative_binomial_distribution

        Examples
        --------
        Draw samples from the distribution:

        A real world example. A company drills wild-cat oil
        exploration wells, each with an estimated probability of
        success of 0.1.  What is the probability of having one success
        for each successive well, that is what is the probability of a
        single success after drilling 5 wells, after 6 wells, etc.?

        >>> s = mkl_random.negative_binomial(1, 0.1, 100000)
        >>> for i in range(1, 11):
        ...    probability = sum(s<i) / 100000.
        ...    print i, "wells drilled, probability of one success =", probability

        RandomStatebytes_ptrRandomState._rand_int16cov must be 2 dimensional and squareRandomState.standard_normal (line 2366)RandomState.gumbel (line 3923)scale <= 0.0mnarrdisableRandomState.binomial (line 4534)RandomState.binomial__exit__Failed to import NumPy_method_alias_dict_gaussian_shortstateshuffleprodPOISNORM
        normal(loc=0.0, scale=1.0, size=None, method='ICDF')

        Draw random samples from a normal (Gaussian) distribution.

        The probability density function of the normal distribution, first
        derived by De Moivre and 200 years later by both Gauss and Laplace
        independently [2]_, is often called the bell curve because of
        its characteristic shape (see the example below).

        The normal distributions occurs often in nature.  For example, it
        describes the commonly occurring distribution of samples influenced
        by a large number of tiny, random disturbances, each with its own
        unique distribution [2]_.

        Parameters
        ----------
        loc : float
            Mean ("centre") of the distribution.
        scale : float
            Standard deviation (spread or "width") of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', 'BoxMuller2', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER, mkl_random.BOXMULLER2

        See Also
        --------
        scipy.stats.distributions.norm : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Gaussian distribution is

        .. math:: p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
                         e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },

        where :math:`\mu` is the mean and :math:`\sigma` the standard
        deviation. The square of the standard deviation, :math:`\sigma^2`,
        is called the variance.

        The function has its peak at the mean, and its "spread" increases with
        the standard deviation (the function reaches 0.607 times its maximum at
        :math:`x + \sigma` and :math:`x - \sigma` [2]_).  This implies that
        `numpy.random.normal` is more likely to return samples lying close to
        the mean, rather than those far away.

        References
        ----------
        .. [1] Wikipedia, "Normal distribution",
               http://en.wikipedia.org/wiki/Normal_distribution
        .. [2] P. R. Peebles Jr., "Central Limit Theorem" in "Probability,
               Random Variables and Random Signal Principles", 4th ed., 2001,
               pp. 51, 51, 125.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, sigma = 0, 0.1 # mean and standard deviation
        >>> s = mkl_random.normal(mu, sigma, 1000)

        Verify the mean and the variance:

        >>> abs(mu - np.mean(s)) < 0.01
        True

        >>> abs(sigma - np.std(s, ddof=1)) < 0.01
        True

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 30, normed=True)
        >>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
        ...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
        ...          linewidth=2, color='r')
        >>> plt.show()

        onbadsigma <= 0.0RandomState.waldnumpy._core.multiarray failed to importasyncio.coroutinesMT2203warndummy_threadingfsigmaBox-Muller2mkl_random.mklrandstridesCannot take a larger sample than population when 'replace=False'low is out of bounds for intp
        tomaxint(size=None)

        Return a sample of uniformly distributed random integers in the interval
        [0, ``np.iinfo("long").max``].

        Parameters
        ----------
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : ndarray
            Drawn samples, with shape `size`.

        See Also
        --------
        randint : Uniform sampling over a given half-open interval of integers.
        random_integers : Uniform sampling over a given closed interval of
            integers.

        Examples
        --------
        >>> RS = np.random_random_intel.RandomState() # need a RandomState object
        >>> RS.tomaxint((2,2,2))
        array([[[1170048599, 1600360186],
                [ 739731006, 1947757578]],
               [[1871712945,  752307660],
                [1601631370, 1479324245]]])
        >>> import sys
        >>> sys.maxint
        2147483647
        >>> RS.tomaxint((2,2,2)) < sys.maxint
        array([[[ True,  True],
                [ True,  True]],
               [[ True,  True],
                [ True,  True]]], dtype=bool)

        __name__flat_found
        choice(a, size=None, replace=True, p=None)

        Generates a random sample from a given 1-D array

                .. versionadded:: 1.7.0

        Parameters
        -----------
        a : 1-D array-like or int
            If an ndarray, a random sample is generated from its elements.
            If an int, the random sample is generated as if a was np.arange(n)
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        replace : boolean, optional
            Whether the sample is with or without replacement
        p : 1-D array-like, optional
            The probabilities associated with each entry in a.
            If not given the sample assumes a uniform distribution over all
            entries in a.

        Returns
        --------
        samples : 1-D ndarray, shape (size,)
            The generated random samples

        Raises
        -------
        ValueError
            If a is an int and less than zero, if a or p are not 1-dimensional,
            if a is an array-like of size 0, if p is not a vector of
            probabilities, if a and p have different lengths, or if
            replace=False and the sample size is greater than the population
            size

        See Also
        ---------
        randint, shuffle, permutation

        Examples
        ---------
        Generate a uniform random sample from np.arange(5) of size 3:

        >>> mkl_random.choice(5, 3)
        array([0, 3, 4])
        >>> #This is equivalent to mkl_random.randint(0,5,3)

        Generate a non-uniform random sample from np.arange(5) of size 3:

        >>> mkl_random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
        array([3, 3, 0])

        Generate a uniform random sample from np.arange(5) of size 3 without
        replacement:

        >>> mkl_random.choice(5, 3, replace=False)
        array([3,1,0])
        >>> #This is equivalent to mkl_random.permutation(np.arange(5))[:3]

        Generate a non-uniform random sample from np.arange(5) of size
        3 without replacement:

        >>> mkl_random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
        array([2, 3, 0])

        Any of the above can be repeated with an arbitrary array-like
        instead of just integers. For instance:

        >>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
        >>> mkl_random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
        array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'],
              dtype='|S11')

        random_integersRandomState.permutationissubdtypendimp < 0RandomState.negative_binomiallngood
        rand(d0, d1, ..., dn)

        Random values in a given shape.

        Create an array of the given shape and propagate it with
        random samples from a uniform distribution
        over ``[0, 1)``.

        Parameters
        ----------
        d0, d1, ..., dn : int, optional
            The dimensions of the returned array, should all be positive.
            If no argument is given a single Python float is returned.

        Returns
        -------
        out : ndarray, shape ``(d0, d1, ..., dn)``
            Random values.

        See Also
        --------
        random

        Notes
        -----
        This is a convenience function. If you want an interface that
        takes a shape-tuple as the first argument, refer to
        mkl_random.random_sample .

        Examples
        --------
        >>> mkl_random.rand(3,2)
        array([[ 0.14022471,  0.96360618],  #random
               [ 0.37601032,  0.25528411],  #random
               [ 0.49313049,  0.94909878]]) #random

        omodetempformat
        bytes(length)

        Return random bytes.

        Parameters
        ----------
        length : int
            Number of random bytes.

        Returns
        -------
        out : str
            String of length `length`.

        Examples
        --------
        >>> mkl_random.bytes(10)
        ' eh\x85\x022SZ\xbf\xa4' #random

        NONDETERMINISTICrayleighinvaccmultingreater_equalreturn_indexsortfloc
        _rand_uint16(low, high, size)

        See `_rand_int32` for documentation, only the return type changes.

        RandomState.standard_gamma (line 2675)RandomState.gumbel
        lognormal(mean=0.0, sigma=1.0, size=None, method='ICDF')

        Draw samples from a log-normal distribution.

        Draw samples from a log-normal distribution with specified mean,
        standard deviation, and array shape.  Note that the mean and standard
        deviation are not the values for the distribution itself, but of the
        underlying normal distribution it is derived from.

        Parameters
        ----------
        mean : float
            Mean value of the underlying normal distribution
        sigma : float, > 0.
            Standard deviation of the underlying normal distribution
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.
        method : 'ICDF, 'BoxMuller', optional
            Sampling method used by Intel MKL. Can also be specified using
            tokens mkl_random.ICDF, mkl_random.BOXMULLER

        Returns
        -------
        samples : ndarray or float
            The desired samples. An array of the same shape as `size` if given,
            if `size` is None a float is returned.

        See Also
        --------
        scipy.stats.lognorm : probability density function, distribution,
            cumulative density function, etc.

        Notes
        -----
        A variable `x` has a log-normal distribution if `log(x)` is normally
        distributed.  The probability density function for the log-normal
        distribution is:

        .. math:: p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
                         e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}

        where :math:`\mu` is the mean and :math:`\sigma` is the standard
        deviation of the normally distributed logarithm of the variable.
        A log-normal distribution results if a random variable is the *product*
        of a large number of independent, identically-distributed variables in
        the same way that a normal distribution results if the variable is the
        *sum* of a large number of independent, identically-distributed
        variables.

        References
        ----------
        .. [1] Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal
               Distributions across the Sciences: Keys and Clues,"
               BioScience, Vol. 51, No. 5, May, 2001.
               http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
        .. [2] Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme
               Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, sigma = 3., 1. # mean and standard deviation
        >>> s = mkl_random.lognormal(mu, sigma, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 100, normed=True, align='mid')

        >>> x = np.linspace(min(bins), max(bins), 10000)
        >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
        ...        / (x * sigma * np.sqrt(2 * np.pi)))

        >>> plt.plot(x, pdf, linewidth=2, color='r')
        >>> plt.axis('tight')
        >>> plt.show()

        Demonstrate that taking the products of random samples from a uniform
        distribution can be fit well by a log-normal probability density
        function.

        >>> # Generate a thousand samples: each is the product of 100 random
        >>> # values, drawn from a normal distribution.
        >>> b = []
        >>> for i in range(1000):
        ...    a = 10. + mkl_random.random(100)
        ...    b.append(np.product(a))

        >>> b = np.array(b) / np.min(b) # scale values to be positive
        >>> count, bins, ignored = plt.hist(b, 100, normed=True, align='mid')
        >>> sigma = np.std(np.log(b))
        >>> mu = np.mean(np.log(b))

        >>> x = np.linspace(min(bins), max(bins), 10000)
        >>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
        ...        / (x * sigma * np.sqrt(2 * np.pi)))

        >>> plt.plot(x, pdf, color='r', linewidth=2)
        >>> plt.show()

        randfunchigh is out of bounds for _randint_typeRandomState.bytes_rand_uint16
        logistic(loc=0.0, scale=1.0, size=None)

        Draw samples from a logistic distribution.

        Samples are drawn from a logistic distribution with specified
        parameters, loc (location or mean, also median), and scale (>0).

        Parameters
        ----------
        loc : float

        scale : float > 0.

        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
                  where the values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.logistic : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Logistic distribution is

        .. math:: P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},

        where :math:`\mu` = location and :math:`s` = scale.

        The Logistic distribution is used in Extreme Value problems where it
        can act as a mixture of Gumbel distributions, in Epidemiology, and by
        the World Chess Federation (FIDE) where it is used in the Elo ranking
        system, assuming the performance of each player is a logistically
        distributed random variable.

        References
        ----------
        .. [1] Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of
               Extreme Values, from Insurance, Finance, Hydrology and Other
               Fields," Birkhauser Verlag, Basel, pp 132-133.
        .. [2] Weisstein, Eric W. "Logistic Distribution." From
               MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/LogisticDistribution.html
        .. [3] Wikipedia, "Logistic-distribution",
               http://en.wikipedia.org/wiki/Logistic_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> loc, scale = 10, 1
        >>> s = mkl_random.logistic(loc, scale, 10000)
        >>> count, bins, ignored = plt.hist(s, bins=50)

        #   plot against distribution

        >>> def logist(x, loc, scale):
        ...     return exp((loc-x)/scale)/(scale*(1+exp((loc-x)/scale))**2)
        >>> plt.plot(bins, logist(bins, loc, scale)*count.max()/\
        ... logist(bins, loc, scale).max())
        >>> plt.show()

        normalleapfrog
        standard_t(df, size=None)

        Draw samples from a standard Student's t distribution with `df` degrees
        of freedom.

        A special case of the hyperbolic distribution.  As `df` gets
        large, the result resembles that of the standard normal
        distribution (`standard_normal`).

        Parameters
        ----------
        df : int
            Degrees of freedom, should be > 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            Drawn samples.

        Notes
        -----
        The probability density function for the t distribution is

        .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
                  \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}

        The t test is based on an assumption that the data come from a
        Normal distribution. The t test provides a way to test whether
        the sample mean (that is the mean calculated from the data) is
        a good estimate of the true mean.

        The derivation of the t-distribution was first published in
        1908 by William Gisset while working for the Guinness Brewery
        in Dublin. Due to proprietary issues, he had to publish under
        a pseudonym, and so he used the name Student.

        References
        ----------
        .. [1] Dalgaard, Peter, "Introductory Statistics With R",
               Springer, 2002.
        .. [2] Wikipedia, "Student's t-distribution"
               http://en.wikipedia.org/wiki/Student's_t-distribution

        Examples
        --------
        From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
        women in Kj is:

        >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
        ...                    7515, 8230, 8770])

        Does their energy intake deviate systematically from the recommended
        value of 7725 kJ?

        We have 10 degrees of freedom, so is the sample mean within 95% of the
        recommended value?

        >>> s = mkl_random.standard_t(10, size=100000)
        >>> np.mean(intake)
        6753.636363636364
        >>> intake.std(ddof=1)
        1142.1232221373727

        Calculate the t statistic, setting the ddof parameter to the unbiased
        value so the divisor in the standard deviation will be degrees of
        freedom, N-1.

        >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
        >>> import matplotlib.pyplot as plt
        >>> h = plt.hist(s, bins=100, normed=True)

        For a one-sided t-test, how far out in the distribution does the t
        statistic appear?

        >>> np.sum(s<t) / float(len(s))
        0.0090699999999999999  #random

        So the p-value is about 0.009, which says the null hypothesis has a
        probability of about 99% of being true.

        leftRandomState.chisquare (line 3057)dtype
        binomial(n, p, size=None)

        Draw samples from a binomial distribution.

        Samples are drawn from a binomial distribution with specified
        parameters, n trials and p probability of success where
        n an integer >= 0 and p is in the interval [0,1]. (n may be
        input as a float, but it is truncated to an integer in use)

        Parameters
        ----------
        n : float (but truncated to an integer)
                parameter, >= 0.
        p : float
                parameter, >= 0 and <=1.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
                  where the values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.binom : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the binomial distribution is

        .. math:: P(N) = \binom{n}{N}p^N(1-p)^{n-N},

        where :math:`n` is the number of trials, :math:`p` is the probability
        of success, and :math:`N` is the number of successes.

        When estimating the standard error of a proportion in a population by
        using a random sample, the normal distribution works well unless the
        product p*n <=5, where p = population proportion estimate, and n =
        number of samples, in which case the binomial distribution is used
        instead. For example, a sample of 15 people shows 4 who are left
        handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4,
        so the binomial distribution should be used in this case.

        References
        ----------
        .. [1] Dalgaard, Peter, "Introductory Statistics with R",
               Springer-Verlag, 2002.
        .. [2] Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill,
               Fifth Edition, 2002.
        .. [3] Lentner, Marvin, "Elementary Applied Statistics", Bogden
               and Quigley, 1972.
        .. [4] Weisstein, Eric W. "Binomial Distribution." From MathWorld--A
               Wolfram Web Resource.
               http://mathworld.wolfram.com/BinomialDistribution.html
        .. [5] Wikipedia, "Binomial-distribution",
               http://en.wikipedia.org/wiki/Binomial_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> n, p = 10, .5  # number of trials, probability of each trial
        >>> s = mkl_random.binomial(n, p, 1000)
        # result of flipping a coin 10 times, tested 1000 times.

        A real world example. A company drills 9 wild-cat oil exploration
        wells, each with an estimated probability of success of 0.1. All nine
        wells fail. What is the probability of that happening?

        Let's do 20,000 trials of the model, and count the number that
        generate zero positive results.

        >>> sum(mkl_random.binomial(9, 0.1, 20000) == 0)/20000.
        # answer = 0.38885, or 38%.

        orightongooddfnumRandomState.random_integers
        random_integers(low, high=None, size=None)

        Random integers of type np.int between `low` and `high`, inclusive.

        Return random integers of type np.int from the "discrete uniform"
        distribution in the closed interval [`low`, `high`].  If `high` is
        None (the default), then results are from [1, `low`]. The np.int
        type translates to the C long type used by Python 2 for "short"
        integers and its precision is platform dependent.

        This function has been deprecated. Use randint instead.

        .. deprecated:: 1.11.0

        Parameters
        ----------
        low : int
            Lowest (signed) integer to be drawn from the distribution (unless
            ``high=None``, in which case this parameter is the *highest* such
            integer).
        high : int, optional
            If provided, the largest (signed) integer to be drawn from the
            distribution (see above for behavior if ``high=None``).
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        out : int or ndarray of ints
            `size`-shaped array of random integers from the appropriate
            distribution, or a single such random int if `size` not provided.

        See Also
        --------
        random.randint : Similar to `random_integers`, only for the half-open
            interval [`low`, `high`), and 0 is the lowest value if `high` is
            omitted.

        Notes
        -----
        To sample from N evenly spaced floating-point numbers between a and b,
        use::

          a + (b - a) * (mkl_random.random_integers(N) - 1) / (N - 1.)

        Examples
        --------
        >>> mkl_random.random_integers(5)
        4
        >>> type(mkl_random.random_integers(5))
        <type 'int'>
        >>> mkl_random.random_integers(5, size=(3.,2.))
        array([[5, 4],
               [3, 3],
               [4, 5]])

        Choose five random numbers from the set of five evenly-spaced
        numbers between 0 and 2.5, inclusive (*i.e.*, from the set
        :math:`{0, 5/8, 10/8, 15/8, 20/8}`):

        >>> 2.5 * (mkl_random.random_integers(5, size=(5,)) - 1) / 4.
        array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ])

        Roll two six sided dice 1000 times and sum the results:

        >>> d1 = mkl_random.random_integers(1, 6, 1000)
        >>> d2 = mkl_random.random_integers(1, 6, 1000)
        >>> dsums = d1 + d2

        Display results as a histogram:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(dsums, 11, normed=True)
        >>> plt.show()

        a must be non-emptyRange exceeds valid bounds
        gumbel(loc=0.0, scale=1.0, size=None)

        Draw samples from a Gumbel distribution.

        Draw samples from a Gumbel distribution with specified location and
        scale.  For more information on the Gumbel distribution, see
        Notes and References below.

        Parameters
        ----------
        loc : float
            The location of the mode of the distribution.
        scale : float
            The scale parameter of the distribution.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar

        See Also
        --------
        scipy.stats.gumbel_l
        scipy.stats.gumbel_r
        scipy.stats.genextreme
        weibull

        Notes
        -----
        The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme
        Value Type I) distribution is one of a class of Generalized Extreme
        Value (GEV) distributions used in modeling extreme value problems.
        The Gumbel is a special case of the Extreme Value Type I distribution
        for maximums from distributions with "exponential-like" tails.

        The probability density for the Gumbel distribution is

        .. math:: p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
                  \beta}},

        where :math:`\mu` is the mode, a location parameter, and
        :math:`\beta` is the scale parameter.

        The Gumbel (named for German mathematician Emil Julius Gumbel) was used
        very early in the hydrology literature, for modeling the occurrence of
        flood events. It is also used for modeling maximum wind speed and
        rainfall rates.  It is a "fat-tailed" distribution - the probability of
        an event in the tail of the distribution is larger than if one used a
        Gaussian, hence the surprisingly frequent occurrence of 100-year
        floods. Floods were initially modeled as a Gaussian process, which
        underestimated the frequency of extreme events.

        It is one of a class of extreme value distributions, the Generalized
        Extreme Value (GEV) distributions, which also includes the Weibull and
        Frechet.

        The function has a mean of :math:`\mu + 0.57721\beta` and a variance
        of :math:`\frac{\pi^2}{6}\beta^2`.

        References
        ----------
        .. [1] Gumbel, E. J., "Statistics of Extremes,"
               New York: Columbia University Press, 1958.
        .. [2] Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme
               Values from Insurance, Finance, Hydrology and Other Fields,"
               Basel: Birkhauser Verlag, 2001.

        Examples
        --------
        Draw samples from the distribution:

        >>> mu, beta = 0, 0.1 # location and scale
        >>> s = mkl_random.gumbel(mu, beta, 1000)

        Display the histogram of the samples, along with
        the probability density function:

        >>> import matplotlib.pyplot as plt
        >>> count, bins, ignored = plt.hist(s, 30, normed=True)
        >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
        ...          * np.exp( -np.exp( -(bins - mu) /beta) ),
        ...          linewidth=2, color='r')
        >>> plt.show()

        Show how an extreme value distribution can arise from a Gaussian process
        and compare to a Gaussian:

        >>> means = []
        >>> maxima = []
        >>> for i in range(0,1000) :
        ...    a = mkl_random.normal(mu, beta, 1000)
        ...    means.append(a.mean())
        ...    maxima.append(a.max())
        >>> count, bins, ignored = plt.hist(maxima, 30, normed=True)
        >>> beta = np.std(maxima) * np.sqrt(6) / np.pi
        >>> mu = np.mean(maxima) - 0.57721*beta
        >>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
        ...          * np.exp(-np.exp(-(bins - mu)/beta)),
        ...          linewidth=2, color='r')
        >>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
        ...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
        ...          linewidth=2, color='g')
        >>> plt.show()

        
        hypergeometric(ngood, nbad, nsample, size=None)

        Draw samples from a Hypergeometric distribution.

        Samples are drawn from a hypergeometric distribution with specified
        parameters, ngood (ways to make a good selection), nbad (ways to make
        a bad selection), and nsample = number of items sampled, which is less
        than or equal to the sum ngood + nbad.

        Parameters
        ----------
        ngood : int or array_like
            Number of ways to make a good selection.  Must be nonnegative.
        nbad : int or array_like
            Number of ways to make a bad selection.  Must be nonnegative.
        nsample : int or array_like
            Number of items sampled.  Must be at least 1 and at most
            ``ngood + nbad``.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(d1, d2, d3)``, then
            ``d1 * d2 * d3`` samples are drawn.  Default is None, in which case a
            single value is returned.

        Returns
        -------
        samples : ndarray or scalar
            The values are all integers in  [0, n].

        See Also
        --------
        scipy.stats.distributions.hypergeom : probability density function,
            distribution or cumulative density function, etc.

        Notes
        -----
        The probability density for the Hypergeometric distribution is

        .. math:: P(x) = \frac{\binom{m}{x}\binom{N-m}{n-x}}{\binom{N}{n}},

        where :math:`0 \le x \le m` and :math:`n+m-N \le x \le n`

        for P(x) the probability of x successes, m = ngood, N = ngood + nbad, and
        n = number of samples.

        Consider an urn with black and white marbles in it, ngood of them
        black and nbad are white. If you draw nsample balls without
        replacement, then the hypergeometric distribution describes the
        distribution of black balls in the drawn sample.

        Note that this distribution is very similar to the binomial
        distribution, except that in this case, samples are drawn without
        replacement, whereas in the Binomial case samples are drawn with
        replacement (or the sample space is infinite). As the sample space
        becomes large, this distribution approaches the binomial.

        References
        ----------
        .. [1] Lentner, Marvin, "Elementary Applied Statistics", Bogden
               and Quigley, 1972.
        .. [2] Weisstein, Eric W. "Hypergeometric Distribution." From
               MathWorld--A Wolfram Web Resource.
               http://mathworld.wolfram.com/HypergeometricDistribution.html
        .. [3] Wikipedia, "Hypergeometric-distribution",
               http://en.wikipedia.org/wiki/Hypergeometric_distribution

        Examples
        --------
        Draw samples from the distribution:

        >>> ngood, nbad, nsamp = 100, 2, 10
        # number of good, number of bad, and number of samples
        >>> s = mkl_random.hypergeometric(ngood, nbad, nsamp, 1000)
        >>> hist(s)
        #   note that it is very unlikely to grab both bad items

        Suppose you have an urn with 15 white and 15 black marbles.
        If you pull 15 marbles at random, how likely is it that
        12 or more of them are one color?

        >>> s = mkl_random.hypergeometric(15, 15, 15, 100000)
        >>> sum(s>=12)/100000. + sum(s<=3)/100000.
        #   answer = 0.003 ... pretty unlikely!

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