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o

�8Va|5�@s:ddlmZddlmZmZmZmZmZmZm	Z	m
Z
mZmZddl
mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z gd�Z!d#dd�d	d
�Z"d$dd�Z#d$d
d�Z$e$Z%d$dd�Z&d$dd�Z'd$dd�Z(d$dd�dd�Z)d$dd�Z*d$dd�Z+d$dd�Z,d$dd�Z-d%dd �Z.d$d!d"�Z/eZ0eZ1e&Z2dS)&�)�	FiniteSet)
�sqrt�log�exp�FallingFactorial�Rational�Eq�Dummy�piecewise_fold�solveset�Integral�)�probability�expectation�density�where�given�pspace�cdf�PSpace�characteristic_function�sample�sample_iter�random_symbols�independent�	dependent�sampling_density�moment_generating_function�quantile�	is_random�sample_stochastic_process)�P�E�Hrrrrrrrr�variance�std�skewness�kurtosis�
covariancer�entropy�medianrr�correlation�factorial_moment�moment�cmomentrr�smomentrr NT)�evaluatecKs6ddlm}|r|||||���S|||||��t�S)a[
    Return the nth moment of a random expression about c.

    .. math::
        moment(X, c, n) = E((X-c)^{n})

    Default value of c is 0.

    Examples
    ========

    >>> from sympy.stats import Die, moment, E
    >>> X = Die('X', 6)
    >>> moment(X, 1, 6)
    -5/2
    >>> moment(X, 2)
    91/6
    >>> moment(X, 1) == E(X)
    True
    r)�Moment)� sympy.stats.symbolic_probabilityr1�doit�rewriter)�X�n�c�	conditionr0�kwargsr1�r:�:/usr/lib/python3/dist-packages/sympy/stats/rv_interface.pyr-sr-cKs@t|�rt|�t�krddlm}|||�St|d|fi|��S)a�
    Variance of a random expression.

    .. math::
        variance(X) = E((X-E(X))^{2})

    Examples
    ========

    >>> from sympy.stats import Die, Bernoulli, variance
    >>> from sympy import simplify, Symbol

    >>> X = Die('X', 6)
    >>> p = Symbol('p')
    >>> B = Bernoulli('B', p, 1, 0)

    >>> variance(2*X)
    35/3

    >>> simplify(variance(B))
    p*(1 - p)
    r)�Variance�)rrrr2r<r.)r5r8r9r<r:r:r;r$.s
r$cKstt||fi|���S)aK
    Standard Deviation of a random expression

    .. math::
        std(X) = \sqrt(E((X-E(X))^{2}))

    Examples
    ========

    >>> from sympy.stats import Bernoulli, std
    >>> from sympy import Symbol, simplify

    >>> p = Symbol('p')
    >>> B = Bernoulli('B', p, 1, 0)

    >>> simplify(std(B))
    sqrt(p*(1 - p))
    )rr$�r5r8r9r:r:r;�standard_deviationLsr?csZt||fi|��}|�dtd���t|t�r#t�fdd�|��D��Stt||����S)an
    Calculuates entropy of a probability distribution.

    Parameters
    ==========

    expression : the random expression whose entropy is to be calculated
    condition : optional, to specify conditions on random expression
    b: base of the logarithm, optional
       By default, it is taken as Euler's number

    Returns
    =======

    result : Entropy of the expression, a constant

    Examples
    ========

    >>> from sympy.stats import Normal, Die, entropy
    >>> X = Normal('X', 0, 1)
    >>> entropy(X)
    log(2)/2 + 1/2 + log(pi)/2

    >>> D = Die('D', 4)
    >>> entropy(D)
    log(4)

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Entropy_(information_theory)
    .. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf
    .. [3] http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf
    �br
csg|]
}|t|���qSr:)r)�.0�prob��baser:r;�
<listcomp>�szentropy.<locals>.<listcomp>)	r�getr�
isinstance�dict�sum�valuesrr)�exprr8r9�pdfr:rCr;r)bs
$
r)cKs~t|�rt|�t�kst|�r"t|�t�kr"ddlm}||||�St|t||fi|��|t||fi|��|fi|��S)aR
    Covariance of two random expressions.

    Explanation
    ===========

    The expectation that the two variables will rise and fall together

    .. math::
        covariance(X,Y) = E((X-E(X)) (Y-E(Y)))

    Examples
    ========

    >>> from sympy.stats import Exponential, covariance
    >>> from sympy import Symbol

    >>> rate = Symbol('lambda', positive=True, real=True, finite=True)
    >>> X = Exponential('X', rate)
    >>> Y = Exponential('Y', rate)

    >>> covariance(X, X)
    lambda**(-2)
    >>> covariance(X, Y)
    0
    >>> covariance(X, Y + rate*X)
    1/lambda
    r)�
Covariance)rrrr2rMr)r5�Yr8r9rMr:r:r;r(�s,���r(cKs8t|||fi|��t||fi|��t||fi|��S)a�
    Correlation of two random expressions, also known as correlation
    coefficient or Pearson's correlation.

    Explanation
    ===========

    The normalized expectation that the two variables will rise
    and fall together

    .. math::
        correlation(X,Y) = E((X-E(X))(Y-E(Y)) / (\sigma_x  \sigma_y))

    Examples
    ========

    >>> from sympy.stats import Exponential, correlation
    >>> from sympy import Symbol

    >>> rate = Symbol('lambda', positive=True, real=True, finite=True)
    >>> X = Exponential('X', rate)
    >>> Y = Exponential('Y', rate)

    >>> correlation(X, X)
    1
    >>> correlation(X, Y)
    0
    >>> correlation(X, Y + rate*X)
    1/sqrt(1 + lambda**(-2))
    )r(r%)r5rNr8r9r:r:r;r+�s"�r+cKs2ddlm}|r||||���S||||��t�S)a\
    Return the nth central moment of a random expression about its mean.

    .. math::
        cmoment(X, n) = E((X - E(X))^{n})

    Examples
    ========

    >>> from sympy.stats import Die, cmoment, variance
    >>> X = Die('X', 6)
    >>> cmoment(X, 3)
    0
    >>> cmoment(X, 2)
    35/12
    >>> cmoment(X, 2) == variance(X)
    True
    r)�
CentralMoment)r2rOr3r4r)r5r6r8r0r9rOr:r:r;r.�sr.cKs2t||fi|��}d||t|||fi|��S)a�
    Return the nth Standardized moment of a random expression.

    .. math::
        smoment(X, n) = E(((X - \mu)/\sigma_X)^{n})

    Examples
    ========

    >>> from sympy.stats import skewness, Exponential, smoment
    >>> from sympy import Symbol
    >>> rate = Symbol('lambda', positive=True, real=True, finite=True)
    >>> Y = Exponential('Y', rate)
    >>> smoment(Y, 4)
    9
    >>> smoment(Y, 4) == smoment(3*Y, 4)
    True
    >>> smoment(Y, 3) == skewness(Y)
    True
    r
)r%r.)r5r6r8r9�sigmar:r:r;r/�s r/cK�t|dfd|i|��S)aN
    Measure of the asymmetry of the probability distribution.

    Explanation
    ===========

    Positive skew indicates that most of the values lie to the right of
    the mean.

    .. math::
        skewness(X) = E(((X - E(X))/\sigma_X)^{3})

    Parameters
    ==========

    condition : Expr containing RandomSymbols
            A conditional expression. skewness(X, X>0) is skewness of X given X > 0

    Examples
    ========

    >>> from sympy.stats import skewness, Exponential, Normal
    >>> from sympy import Symbol
    >>> X = Normal('X', 0, 1)
    >>> skewness(X)
    0
    >>> skewness(X, X > 0) # find skewness given X > 0
    (-sqrt(2)/sqrt(pi) + 4*sqrt(2)/pi**(3/2))/(1 - 2/pi)**(3/2)

    >>> rate = Symbol('lambda', positive=True, real=True, finite=True)
    >>> Y = Exponential('Y', rate)
    >>> skewness(Y)
    2
    �r8�r/r>r:r:r;r&s#r&cKrQ)a'
    Characterizes the tails/outliers of a probability distribution.

    Explanation
    ===========

    Kurtosis of any univariate normal distribution is 3. Kurtosis less than
    3 means that the distribution produces fewer and less extreme outliers
    than the normal distribution.

    .. math::
        kurtosis(X) = E(((X - E(X))/\sigma_X)^{4})

    Parameters
    ==========

    condition : Expr containing RandomSymbols
            A conditional expression. kurtosis(X, X>0) is kurtosis of X given X > 0

    Examples
    ========

    >>> from sympy.stats import kurtosis, Exponential, Normal
    >>> from sympy import Symbol
    >>> X = Normal('X', 0, 1)
    >>> kurtosis(X)
    3
    >>> kurtosis(X, X > 0) # find kurtosis given X > 0
    (-4/pi - 12/pi**2 + 3)/(1 - 2/pi)**2

    >>> rate = Symbol('lamda', positive=True, real=True, finite=True)
    >>> Y = Exponential('Y', rate)
    >>> kurtosis(Y)
    9

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Kurtosis
    .. [2] http://mathworld.wolfram.com/Kurtosis.html
    �r8rSr>r:r:r;r',s*r'cKstt||�fd|i|��S)a�
    The factorial moment is a mathematical quantity defined as the expectation
    or average of the falling factorial of a random variable.

    .. math::
        factorial-moment(X, n) = E(X(X - 1)(X - 2)...(X - n + 1))

    Parameters
    ==========

    n: A natural number, n-th factorial moment.

    condition : Expr containing RandomSymbols
            A conditional expression.

    Examples
    ========

    >>> from sympy.stats import factorial_moment, Poisson, Binomial
    >>> from sympy import Symbol, S
    >>> lamda = Symbol('lamda')
    >>> X = Poisson('X', lamda)
    >>> factorial_moment(X, 2)
    lamda**2
    >>> Y = Binomial('Y', 2, S.Half)
    >>> factorial_moment(Y, 2)
    1/2
    >>> factorial_moment(Y, 2, Y > 1) # find factorial moment for Y > 1
    2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Factorial_moment
    .. [2] http://mathworld.wolfram.com/FactorialMoment.html
    r8)rr)r5r6r8r9r:r:r;r,Ys%r,cKst|�s|Sddlm}ddlm}ddlm}tt|�|�rTt|��	|�}g}|�
�D]#\}}	|	tdd�krOd|	t|��t
||��tdd�krO|�|�q,t|�Stt|�|�sbtt|�|�r�t|��	|�}td�}
tt||
�tdd��|
t|�j�}|Stdtt|����)	aN
    Calculuates the median of the probability distribution.

    Explanation
    ===========

    Mathematically, median of Probability distribution is defined as all those
    values of `m` for which the following condition is satisfied

    .. math::
        P(X\leq m) \geq  \frac{1}{2} \text{ and} \text{ } P(X\geq m)\geq \frac{1}{2}

    Parameters
    ==========

    X: The random expression whose median is to be calculated.

    Returns
    =======

    The FiniteSet or an Interval which contains the median of the
    random expression.

    Examples
    ========

    >>> from sympy.stats import Normal, Die, median
    >>> N = Normal('N', 3, 1)
    >>> median(N)
    {3}
    >>> D = Die('D')
    >>> median(D)
    {3, 4}

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions

    r)�ContinuousPSpace)�DiscretePSpace)�FinitePSpacer
r=�xz#The median of %s is not implemeted.)r�sympy.stats.crvrU�sympy.stats.drvrV�sympy.stats.frvrWrGr�compute_cdf�itemsrrr�appendrr	rr
�set�NotImplementedError�str)r5r0r9rUrVrWr�result�key�valuerXr:r:r;r*�s.)��
�$r*cKs�t|t||fi|��|t||fi|��|t||fi|��|fi|��}t||fi|��t||fi|��t||fi|��}||S)a%
    Calculates the co-skewness of three random variables.

    Explanation
    ===========

    Mathematically Coskewness is defined as

    .. math::
        coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}}

    Parameters
    ==========

    X : RandomSymbol
            Random Variable used to calculate coskewness
    Y : RandomSymbol
            Random Variable used to calculate coskewness
    Z : RandomSymbol
            Random Variable used to calculate coskewness
    condition : Expr containing RandomSymbols
            A conditional expression

    Examples
    ========

    >>> from sympy.stats import coskewness, Exponential, skewness
    >>> from sympy import symbols
    >>> p = symbols('p', positive=True)
    >>> X = Exponential('X', p)
    >>> Y = Exponential('Y', 2*p)
    >>> coskewness(X, Y, Y)
    0
    >>> coskewness(X, Y + X, Y + 2*X)
    16*sqrt(85)/85
    >>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3)
    9*sqrt(170)/85
    >>> coskewness(Y, Y, Y) == skewness(Y)
    True
    >>> coskewness(X, Y + p*X, Y + 2*p*X)
    4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2)))

    Returns
    =======

    coskewness : The coskewness of the three random variables

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Coskewness

    )rr%)r5rN�Zr8r9�num�denr:r:r;�
coskewness�s6����"�rh)rN)N)T)3�
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rr�rvrrrrrrrrrrrrrrrrrrr �__all__r-r$r?r%r)r(r+r.r/r&r'r,r*rhr!r"r#r:r:r:r;�<module>s*0T	



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