python/mxnet/numpy/random.py - master - githubmirror/incubator-mxnet

python/mxnet/numpy/random.py
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"""Namespace for ops used in imperative programming."""

from ..ndarray import numpy as _mx_nd_np
from ..random import seed
from ..util import wrap_ctx_to_device_func


__all__ = ["randint", "uniform", "normal", "choice", "rand", "multinomial", "multivariate_normal",
           "logistic", "gumbel", "f",
           "laplace",
           "shuffle", "randn", "gamma", "beta", "chisquare", "exponential", "lognormal",
           "weibull", "pareto", "power", "rayleigh",
           "seed"]


@wrap_ctx_to_device_func
def randint(low, high=None, size=None, dtype=None, device=None, out=None):
    r"""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 one above 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'.
    device : Device, optional
        Device context of output. Default is current device.
    out : ndarray, optional
        The output ndarray (default is `None`).

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

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

    >>> np.random.randint(5, size=(2, 4))
    array([[4, 0, 2, 1],
        [3, 2, 2, 0]])
    """
    return _mx_nd_np.random.randint(low, high, size, dtype, device, out)


@wrap_ctx_to_device_func
def uniform(low=0.0, high=1.0, size=None, dtype=None, device=None, out=None):
    r"""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, ndarray, 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, ndarray, optional
        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.  If size is ``None`` (default),
        a scalar tensor containing a single value is returned if
        ``low`` and ``high`` are both scalars. Otherwise,
        ``np.broadcast(low, high).size`` samples are drawn.
    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples.
        When npx.is_np_default_dtype() returns False, default dtype is float32;
        When npx.is_np_default_dtype() returns True, default dtype is float64.
    device : Device, optional
        Device context of output. Default is current device.

    Returns
    -------
    out : ndarray
        Drawn samples from the parameterized uniform distribution.

    See Also
    --------
    randint : Discrete uniform distribution, yielding integers.
    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.

    When ``high`` == ``low``, values of ``low`` will be returned.
    If ``high`` < ``low``, the results are officially undefined
    and may eventually raise an error, i.e. do not rely on this
    function to behave when passed arguments satisfying that
    inequality condition.
    """
    return _mx_nd_np.random.uniform(low, high, size=size, device=device, dtype=dtype, out=out)


@wrap_ctx_to_device_func
def normal(loc=0.0, scale=1.0, size=None, dtype=None, device=None, out=None):
    r"""Draw random samples from a normal (Gaussian) distribution.

    Samples are distributed according to a normal distribution parametrized
    by *loc* (mean) and *scale* (standard deviation).

    Parameters
    ----------
    loc : float, optional
        Mean (centre) of the distribution.
    scale : float, optional
        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. If size is `None` (default), a scalar tensor containing
        a single value is returned if loc and scale are both scalars. Otherwise,
        ``np.broadcast(low, high).size`` samples are drawn.
    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples.
        When npx.is_np_default_dtype() returns False, default dtype is float32;
        When npx.is_np_default_dtype() returns True, default dtype is float64.
    device : Device, optional
        Device context of output, default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray
        Drawn samples from the parameterized `normal distribution` [1]_.

    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",
           https://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
    --------
    >>> mu, sigma = 0, 0.1 # mean and standard deviation
    >>> s = np.random.normal(mu, sigma, 1000)

    Verify the mean and the variance:

    >>> np.abs(mu - np.mean(s)) < 0.01
    array(True)
    """
    return _mx_nd_np.random.normal(loc, scale, size, dtype, device, out)


@wrap_ctx_to_device_func
def lognormal(mean=0.0, sigma=1.0, size=None, dtype=None, device=None, out=None):
    r"""Draw samples from a log-normal distribution.

    Draw samples from a `log-normal distribution` [1]_ 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 or array_like of floats, optional
        Mean value of the underlying normal distribution. Default is 0.
    sigma : float or array_like of floats, optional
        Standard deviation of the underlying normal distribution. Must be
        non-negative. Default is 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.  If size is ``None`` (default),
        a single value is returned if ``mean`` and ``sigma`` are both scalars.
        Otherwise, ``np.broadcast(mean, sigma).size`` samples are drawn.
    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples. Default is 'float32'
    device : Device, optional
        Device context of output. Default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized log-normal distribution.

    Notes
    -----
    A variable `x` has a log-normal distribution if `log(x)` is normally
    distributed.  The `probability density function for the log-normal
    distribution` [2]_ 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://www.statlit.org/pdf/2001-Limpert-Bioscience2.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 = np.random.lognormal(mu, sigma, 1000)
    """
    return _mx_nd_np.random.lognormal(mean, sigma, size, dtype, device, out)


@wrap_ctx_to_device_func
def logistic(loc=0.0, scale=1.0, size=None, device=None, out=None):
    r"""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 or array_like of floats, optional
        Parameter of the distribution. Default is 0.
    scale : float or array_like of floats, optional
        Parameter of the distribution. Must be non-negative.
        Default is 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.  If size is ``None`` (default),
        a single value is returned if ``loc`` and ``scale`` are both scalars.
        Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
    device : Device, optional
        Device context of output, default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized logistic distribution.

    Examples
    --------
    Draw samples from the distribution:
    >>> loc, scale = 10, 1
    >>> s = np.random.logistic(loc, scale, 10000)
    >>> import matplotlib.pyplot as plt
    >>> count, bins, ignored = plt.hist(s, bins=50)
    #   plot against distribution
    >>> def logist(x, loc, scale):
    ...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
    >>> lgst_val = logist(bins, loc, scale)
    >>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
    >>> plt.show()
    """
    return _mx_nd_np.random.logistic(loc, scale, size, device, out)


@wrap_ctx_to_device_func
def gumbel(loc=0.0, scale=1.0, size=None, device=None, out=None):
    r"""Draw samples from a Gumbel distribution.

    Draw samples from a Gumbel distribution with specified location and
    scale.

    Parameters
    ----------
    loc : float or array_like of floats, optional
        The location of the mode of the distribution. Default is 0.
    scale : float or array_like of floats, optional
        The scale parameter of the distribution. Default is 1. Must 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.  If size is ``None`` (default),
        a single value is returned if ``loc`` and ``scale`` are both scalars.
        Otherwise, ``np.broadcast(loc, scale).size`` samples are drawn.
    device : Device, optional
        Device context of output, default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized Gumbel distribution.

    Examples
    --------
    Draw samples from the distribution:
    >>> mu, beta = 0, 0.1 # location and scale
    >>> s = np.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, density=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 = np.random.normal(mu, beta, 1000)
    ...    means.append(a.mean())
    ...    maxima.append(a.max())
    >>> count, bins, ignored = plt.hist(maxima, 30, density=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()
    """
    return _mx_nd_np.random.gumbel(loc, scale, size, device, out)


def multinomial(n, pvals, size=None, **kwargs):
    r"""
    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.
    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
        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 1000 times, and 1000 times again:

    >>> np.random.multinomial(1000, [1/6.]*6, size=2)
    array([[164, 161, 179, 158, 150, 188],
           [178, 162, 177, 143, 163, 177]])

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

    >>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
    array([19, 14, 12, 11, 21, 23])
    >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])
    array([32, 68])
    """
    return _mx_nd_np.random.multinomial(n, pvals, size, **kwargs)


# pylint: disable=unused-argument
def multivariate_normal(mean, cov, size=None, check_valid=None, tol=None):
    """
    multivariate_normal(mean, cov, size=None, check_valid=None, tol=None)

    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.

    This operator is a little different from the one in official NumPy.
    The official NumPy operator only accepts 1-D ndarray as mean and 2-D ndarray as cov,
    whereas the operator in MXNet np supports batch operation and auto-broadcasting.

    Both `mean` and `cov` may have any number of leading dimensions, which correspond
    to a batch shape. They are not necessarily assumed to have the same batch shape,
    just ones which can be broadcasted.

    Parameters
    ----------
    mean : K-D ndarray, of shape (..., N)
        Mean of the N-dimensional distribution.
    cov : (K+1)-D ndarray, of shape (..., N, N)
        Covariance matrix of the distribution. The last two dimensions 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`` identically distributed batchs of samples are
        generated, and packed in an `m`-by-`n`-by-`k` arrangement.
        If no shape is specified, a batch of (`N`-D) sample is returned.
    check_valid : { 'warn', 'raise', 'ignore' }, optional
        Behavior when the covariance matrix is not positive semidefinite.
        (Not supported)
    tol : float, optional
        Tolerance when checking the singular values in covariance matrix.
        cov is cast to double before the check.
        (Not supported)

    Returns
    -------
    out : ndarray
        The input shape of `mean` and `cov` should satisfy the requirements of broadcasting.
        If the parameter `size` is not provided,
        the output shape is ``np.broadcast(mean.shape, cov.shape[:-1])``.
        Otherwise, the output shape is ``size + np.broadcast(mean.shape, cov.shape[:-1])``

    Examples
    --------
    >>> mean = np.array([1, 2])
    >>> cov = np.array([[1, 0], [0, 1]])
    >>> x = np.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] # random

    # Performs autobroadcasting when the batch shape of
    # `mean` and `cov` is different but compatible.

    >>> mean = np.zeros((3,2)) # shape (3, 2)
    >>> cov = np.array([[1, 0], [0, 100]]) # shape (2, 2)
    >>> x = np.random.multivariate_normal(mean, cov)
    >>> x
    array([[-1.6115597 , -8.726251  ],
           [ 2.2425299 ,  2.8104177 ],
           [ 0.36229908, -8.386591  ]])
    """
    return _mx_nd_np.random.multivariate_normal(mean, cov, size=size, check_valid=None, tol=None)


@wrap_ctx_to_device_func
def choice(a, size=None, replace=True, p=None, device=None, out=None):
    r"""Generates a random sample from a given 1-D array

    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 were np.arange(a)
    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.
    device : Device, optional
        Device context of output. Default is current device.

    Returns
    --------
    samples : ndarray
        The generated random samples

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

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

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

    >>> np.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:

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

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

    >>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
    array([2, 3, 0])
    """
    return _mx_nd_np.random.choice(a, size, replace, p, device, out)


@wrap_ctx_to_device_func
def rayleigh(scale=1.0, size=None, device=None, out=None):
    r"""Draw samples from a Rayleigh distribution.

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

    Parameters
    ----------
    scale : float, optional
        Scale, also equals the mode. Must be non-negative. Default is 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.  If size is ``None`` (default),
        a single value is returned if ``scale`` is a scalar.  Otherwise,
        ``np.array(scale).size`` samples are drawn.
    device : Device, optional
        Device context of output, default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized Rayleigh distribution.
    """
    return _mx_nd_np.random.rayleigh(scale, size, device, out)


def rand(*size, **kwargs):
    r"""Random values in a given shape.

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

    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
    -------
    out : ndarray
       Random values.

    Examples
    --------
    >>> np.random.rand(3,2)
    array([[ 0.14022471,  0.96360618],  #random
           [ 0.37601032,  0.25528411],  #random
           [ 0.49313049,  0.94909878]]) #random
    """
    output_shape = ()
    for s in size:
        output_shape += (s,)
    return _mx_nd_np.random.uniform(0, 1, size=output_shape, **kwargs)


@wrap_ctx_to_device_func
def exponential(scale=1.0, size=None, device=None, out=None):
    r"""Draw samples from an exponential distribution.

    Parameters
    ----------
    scale : float or array_like of floats
        The scale parameter, :math:`\beta = 1/\lambda`. Must 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.  If size is ``None`` (default),
        a single value is returned if ``scale`` is a scalar.  Otherwise,
        ``np.array(scale).size`` samples are drawn.
    device : Device, optional
        Device context of output, default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized exponential distribution.
    """
    return _mx_nd_np.random.exponential(scale, size=size, device=device, out=out)


@wrap_ctx_to_device_func
def weibull(a, size=None, device=None, out=None):
    r"""Draw samples from a 1-parameter Weibull distribution with given parameter a
    via inversion.

    Parameters
    ----------
    a : float or array_like of floats
        Shape of the distribution. Must 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.  If size is ``None`` (default),
        a single value is returned if ``a`` is a scalar. Otherwise,
        ``np.array(a).size`` samples are drawn.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the 1-parameter Weibull distribution.

    Examples
    --------
    >>> np.random.weibull(a=5)
    array(0.9553641)
    >>> np.random.weibull(a=5, size=[2,3])
    array([[1.0466299 , 1.1320982 , 0.98415005],
          [1.1430776 , 0.9532727 , 1.1344457 ]])
    >>> np.random.weibull(a=np.array([2,3])
    array([0.98843634, 1.0125613 ])
    The Weibull distribution is one of a class of Generalized Extreme
    Value (GEV) distributions. This class includes the Gumbel and Frechet
    distributions.
    The probability density for the Weibull distribution is
    f(x) = \frac{a}{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},
    where a is the shape and \lambda the scale. The generated 1-parameter Weibull
    sample has the scale parameter \lambda = 1.
    The Weibull distribution is commonly used in reliability engineering to
    model time to failure, in modeling particle sizes, in information retrieval
    to model dwell time on pages, in quantitative finance to model risk etc.
    """
    return _mx_nd_np.random.weibull(a, size=size, device=device, out=out)


@wrap_ctx_to_device_func
def pareto(a, size=None, device=None, out=None):
    r"""Draw samples from a Pareto II or Lomax distribution with specified shape a.

    Parameters
    ----------
    a : float or array_like of floats
            Shape of the distribution. Must 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.  If size is ``None`` (default),
        a single value is returned if ``a`` is a scalar. Otherwise,
        ``np.array(a).size`` samples are drawn.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the Pareto distribution.

    Examples
    --------
    >>> np.random.pareto(a=5)
    array(0.12749612)
    >>> mx.numpy.random.pareto(a=5, size=[2,3])
    array([[0.06933999, 0.0344373 , 0.10654891],
            [0.0311172 , 0.12911797, 0.03370714]])
    >>> np.random.pareto(a=np.array([2,3])
    array([0.26636696, 0.15685666])
    The probability density for the Pareto distribution is f(x) = \frac{am^a}{x^{a+1}}
    where a is the shape and m the scale. Here m is assumed 1. The Pareto distribution
    is a power law distribution. Pareto created it to describe the wealth in the economy.
    """
    return _mx_nd_np.random.pareto(a, size=size, device=device, out=out)


@wrap_ctx_to_device_func
def power(a, size=None, device=None, out=None):
    r"""Draw samples in [0, 1] from a power distribution with given parameter a.

    Parameters
    ----------
    a : float or array_like of floats
        Shape of the distribution. Must 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.  If size is ``None`` (default),
        a single value is returned if ``a`` is a scalar. Otherwise,
        ``np.array(a).size`` samples are drawn.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the power distribution.

    Examples
    --------
    >>> np.random.power(a=5)
    array(0.8602478)
    >>> np.random.power(a=5, size=[2,3])
    array([[0.988391  , 0.5153122 , 0.9383134 ],
           [0.9078098 , 0.87819266, 0.730635]])
    >>> np.random.power(a=np.array([2,3])
    array([0.7499419 , 0.88894516])
    The probability density function is f(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.
    The power distribution is just the inverse of the Pareto distribution and
    a special case of the Beta distribution.
    """
    return _mx_nd_np.random.power(a, size=size, device=device, out=out)


def shuffle(x):
    """
    Modify a sequence in-place by shuffling its contents.

    This function only shuffles the array along the first axis of a
    multi-dimensional array. The order of sub-arrays is changed but
    their contents remain the same.

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

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

    Multi-dimensional arrays are only shuffled along the first axis:

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


@wrap_ctx_to_device_func
def gamma(shape, scale=1.0, size=None, dtype=None, device=None, out=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.

    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.

    Parameters
    ----------
    shape : float or array_like of floats
        The shape of the gamma distribution. Should be greater than zero.
    scale : float or array_like of floats, optional
        The scale of the gamma distribution. Should be greater than zero.
        Default is equal to 1.
    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples. Default is 'float32'.
    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.  If size is ``None`` (default),
        a single value is returned if ``shape`` and ``scale`` are both scalars.
        Otherwise, ``np.broadcast(shape, scale).size`` samples are drawn.
    device : Device, optional
        Device context of output. Default is current device.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized gamma distribution.
    """
    return _mx_nd_np.random.gamma(shape, scale, size, dtype, device, out)


@wrap_ctx_to_device_func
def beta(a, b, size=None, dtype=None, device=None):
    r"""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 or array_like of floats
        Alpha, positive (>0).
    b : float or array_like of floats
        Beta, positive (>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.  If size is ``None`` (default),
        a single value is returned if ``a`` and ``b`` are both scalars.
        Otherwise, ``np.broadcast(a, b).size`` samples are drawn.
    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples. Default is 'float32'.
        Dtype 'float32' or 'float64' is strongly recommended,
        since lower precision might lead to out of range issue.
    device : Device, optional
        Device context of output. Default is current device.

    Notes
    -----
    To use this operator with scalars as input, please run
    ``npx.set_np()`` first.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized beta distribution.
    """
    return _mx_nd_np.random.beta(a, b, size=size, dtype=dtype, device=device)


@wrap_ctx_to_device_func
def f(dfnum, dfden, size=None, device=None):
    r"""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 must 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 or ndarray of floats
        Degrees of freedom in numerator, must be > 0.
    dfden : float or ndarray of float
        Degrees of freedom in denominator, must 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.  If size is ``None`` (default),
        a single value is returned if ``dfnum`` and ``dfden`` are both scalars.
        Otherwise, ``np.broadcast(dfnum, dfden).size`` samples are drawn.
    device : Device, optional
        Device context of output. Default is current device.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized Fisher 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 = np.random.f(dfnum, dfden, 1000)

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

    >>> np.sort(s)[-10]
    7.61988120985 # random

    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.
    """
    return _mx_nd_np.random.f(dfnum, dfden, size=size, device=device)


@wrap_ctx_to_device_func
def chisquare(df, size=None, dtype=None, device=None):
    r"""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 : float or ndarray of floats
         Number of degrees of freedom, must 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.  If size is ``None`` (default),
        a single value is returned if ``df`` is a scalar.  Otherwise,
        ``np.array(df).size`` samples are drawn.
    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples. Default is 'float32'.
    device : Device, optional
        Device context of output. Default is current device.

    Returns
    -------
    out : ndarray or scalar
        Drawn samples from the parameterized `chi-square distribution` [1]_.

    Raises
    ------
    ValueError
        When `df` <= 0 or when an inappropriate `size`
        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"
           https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

    Examples
    --------
    >>> np.random.chisquare(2,4)
    array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random
    """
    return _mx_nd_np.random.chisquare(df, size=size, dtype=dtype, device=device)


def randn(*size, **kwargs):
    r"""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
        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.
    Notes
    -----
    For random samples from :math:`N(\mu, \sigma^2)`, use:
    ``sigma * np.random.randn(...) + mu``
    Examples
    --------
    >>> np.random.randn()
    2.1923875335537315 #random
    Two-by-four array of samples from N(3, 6.25):
    >>> 2.5 * np.random.randn(2, 4) + 3
    array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],  #random
        [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]]) #random
    """
    output_shape = ()
    for s in size:
        output_shape += (s,)
    return _mx_nd_np.random.normal(0, 1, size=output_shape, **kwargs)


@wrap_ctx_to_device_func
def laplace(loc=0.0, scale=1.0, size=None, dtype=None, device=None, out=None):
    r"""Draw random samples from a Laplace distribution.

    Samples are distributed according to a Laplace distribution parametrized
    by *loc* (mean) and *scale* (the exponential decay).

    Parameters
    ----------
    loc : float, The position of the distribution peak.

    scale : float, 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.

    dtype : {'float16', 'float32', 'float64'}, optional
        Data type of output samples. Default is 'float32'
    device : Device, optional
        Device context of output. Default is current device.
    out : ``ndarray``, optional
        Store output to an existing ``ndarray``.

    Returns
    -------
    out : ndarray
        Drawn samples from the parameterized Laplace distribution.
    """
    return _mx_nd_np.random.laplace(loc, scale, size, dtype, device, out)