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Create a continuous random variable with a log-normal distribution.

The density of the log-normal distribution is given by

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

with :math:`x \geq 0`.
(more...)

        def LogNormal(name, mean, std):
    r"""
    Create a continuous random variable with a log-normal distribution.

    The density of the log-normal distribution is given by

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

    with :math:`x \geq 0`.

    Parameters
    ==========

    mu : Real number, the log-scale
    sigma : Real number, :math:`\sigma^2 > 0` a shape

    Returns
    =======

    A RandomSymbol.

    Examples
    ========

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

    >>> mu = Symbol("mu", real=True)
    >>> sigma = Symbol("sigma", positive=True)
    >>> z = Symbol("z")

    >>> X = LogNormal("x", mu, sigma)

    >>> D = density(X)(z)
    >>> pprint(D, use_unicode=False)
                          2
           -(-mu + log(z))
           -----------------
                      2
      ___      2*sigma
    \/ 2 *e
    ------------------------
            ____
        2*\/ pi *sigma*z


    >>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1

    >>> density(X)(z)
    sqrt(2)*exp(-log(z)**2/2)/(2*sqrt(pi)*z)

    References
    ==========

    .. [1] http://en.wikipedia.org/wiki/Lognormal
    .. [2] http://mathworld.wolfram.com/LogNormalDistribution.html
    """

    return rv(name, LogNormalDistribution, (mean, std))
        


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,
def test_lognormal():
    mean = Symbol('mu', real=True, bounded=True)
    std = Symbol('sigma', positive=True, real=True, bounded=True)
    X = LogNormal('x', mean, std)
    # The sympy integrator can't do this too well
    # Test sampling: Only e^mean in sample std of 0
    for i in range(3):
        X = LogNormal('x', i, 0)
        assert S(sample(X)) == N(exp(i))
    # The sympy integrator can't do this too well
    sigma = Symbol("sigma", positive=True)
 
    X = LogNormal('x', mu, sigma)
    assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
                                    /(2*sigma**2))/(2*x*sqrt(pi)*sigma))