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Create a Continuous Random Variable with a Benini distribution.

The density of the Benini distribution is given by

.. math::
    f(x) := e^{-\alpha\log{\frac{x}{\sigma}}
            -\beta\log^2\left[{\frac{x}{\sigma}}\right]}
            \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)

This is a heavy-tailed distrubtion and is also known as the log-Rayleigh(more...)

        def Benini(name, alpha, beta, sigma):
    r"""
    Create a Continuous Random Variable with a Benini distribution.

    The density of the Benini distribution is given by

    .. math::
        f(x) := e^{-\alpha\log{\frac{x}{\sigma}}
                -\beta\log^2\left[{\frac{x}{\sigma}}\right]}
                \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)

    This is a heavy-tailed distrubtion and is also known as the log-Rayleigh
    distribution.

    Parameters
    ==========

    alpha : Real number, `\alpha > 0`, a shape
    beta : Real number, `\beta > 0`, a shape
    sigma : Real number, `\sigma > 0`, a scale

    Returns
    =======

    A RandomSymbol.

    Examples
    ========

    >>> from sympy.stats import Benini, density
    >>> from sympy import Symbol, simplify, pprint

    >>> alpha = Symbol("alpha", positive=True)
    >>> beta = Symbol("beta", positive=True)
    >>> sigma = Symbol("sigma", positive=True)
    >>> z = Symbol("z")

    >>> X = Benini("x", alpha, beta, sigma)

    >>> D = density(X)(z)
    >>> pprint(D, use_unicode=False)
    /                  /  z  \\             /  z  \            2/  z  \
    |        2*beta*log|-----||  - alpha*log|-----| - beta*log  |-----|
    |alpha             \sigma/|             \sigma/             \sigma/
    |----- + -----------------|*e
    \  z             z        /

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Benini_distribution
    .. [2] http://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html
    """

    return rv(name, BeniniDistribution, (alpha, beta, sigma))
        


src/s/y/sympy-HEAD/sympy/stats/tests/test_continuous_rv.py   sympy(Download)
from sympy.stats import (P, E, where, density, variance, covariance, skewness,
                         given, pspace, cdf, ContinuousRV, sample,
                         Arcsin, Benini, Beta, BetaPrime, Cauchy,
                         Chi, ChiSquared,
                         ChiNoncentral, Dagum, Erlang, Exponential,
    sigma = Symbol("sigma", positive=True)
 
    X = Benini('x', alpha, b, sigma)
    assert density(X)(x) == ((alpha/x + 2*b*log(x/sigma)/x)
                          *exp(-alpha*log(x/sigma) - b*log(x/sigma)**2))