U Zʹjq6@sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z ddl mZddlmZdd lmZdd lmZd d lmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(d ddddddddddddddddddd d!d"d#d$d%d&d'd(d)d*gZ)d=d,d-d.d$Z*d>d/dZ+d?d0d1Z,e,Z-d@d2dZ.dAd3dZ/dBd4d"Z0dCd,d-d5d%Z1dDd6d(Z2dEd7dZ3dFd8dZ4dGd9d#Z5dHd:dZ6dId;d<Z7eZ8eZ9e.Z:d+S)J) FiniteSet)Rational)Eq)Dummy)FallingFactorial)explog)sqrt)piecewise_fold)Integral)solveset) probability expectationdensitywheregivenpspacecdfPSpacecharacteristic_functionsample sample_iterrandom_symbols independent dependentsampling_densitymoment_generating_functionquantile is_randomsample_stochastic_processPEHrrrrrrrrvariancestdskewnesskurtosis covariancerentropymedianrr correlationfactorial_momentmomentcmomentrrsmomentrr NT)evaluatecKs6ddlm}|r"|||||S|||||tS)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_probabilityr1doitrewriter )Xnc conditionr0kwargsr1r:W/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/stats/rv_interface.pyr-s cKs<t|r,t|tkr,ddlm}|||St|d|f|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$5s  cKstt||f|S)aK Standard Deviation of a random expression .. math:: std(X) = \sqrt(E((X-E(X))^{2})) Examples ======== >>> from sympy.stats import Bernoulli, std >>> from sympy import Symbol, simplify >>> p = Symbol('p') >>> B = Bernoulli('B', p, 1, 0) >>> simplify(std(B)) sqrt(p*(1 - p)) )r r$r5r8r9r:r:r;standard_deviationSsr?c sVt||f|}|dtdt|trBtfdd|DStt|| 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_%28information_theory%29 .. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf .. [3] https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf br c3s|]}| t|VqdS)N)r).0Zprobbaser:r; szentropy..) rgetr isinstancedictsumvaluesrr)exprr8r9Zpdfr:rBr;r)is $ cKsrt|rt|tks,t|rDt|tkrDddlm}||||St|t||f||t||f||f|S)aE 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) >>> 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)rrrr2rKr)r5Yr8r9rKr:r:r;r(s,  cKs,t|||f|t||f|t||f|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) >>> 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%)r5rLr8r9r:r:r;r+s cKs2ddlm}|r ||||S||||tS)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)r2rMr3r4r )r5r6r8r0r9rMr:r:r;r.s cKs*t||f|}d||t|||f|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) >>> 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.)r5r6r8r9sigmar:r:r;r/scKst|dfd|i|S)aA 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) >>> Y = Exponential('Y', rate) >>> skewness(Y) 2 r8r/r>r:r:r;r&s#cKst|dfd|i|S)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) >>> Y = Exponential('Y', rate) >>> kurtosis(Y) 9 References ========== .. [1] https://en.wikipedia.org/wiki/Kurtosis .. [2] https://mathworld.wolfram.com/Kurtosis.html r8rPr>r:r:r;r'3s*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] https://mathworld.wolfram.com/FactorialMoment.html r8)rr)r5r6r8r9r:r:r;r,`s%c Kst|s |Sddlm}ddlm}ddlm}tt||rt| |}g}| D]F\}} | t ddkrXd| t| t ||t ddkrX||qXt|Stt|||frt| |}td} tt|| t dd| t|j}|Stdtt|d S) 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 implemented.N)rZsympy.stats.crvrRZsympy.stats.drvrSZsympy.stats.frvrTrFrZ compute_cdfitemsrrrappendrrr r setNotImplementedErrorstr) r5r0r9rRrSrTrresultkeyvaluerUr:r:r;r*s,)    $cKsrt|t||f||t||f||t||f||f|}t||f|t||f|t||f|}||S)a% Calculates the co-skewness of three random variables. Explanation =========== Mathematically Coskewness is defined as .. math:: coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}} Parameters ========== X : RandomSymbol Random Variable used to calculate coskewness Y : RandomSymbol Random Variable used to calculate coskewness Z : RandomSymbol Random Variable used to calculate coskewness condition : Expr containing RandomSymbols A conditional expression Examples ======== >>> from sympy.stats import coskewness, Exponential, skewness >>> from sympy import symbols >>> p = symbols('p', positive=True) >>> X = Exponential('X', p) >>> Y = Exponential('Y', 2*p) >>> coskewness(X, Y, Y) 0 >>> coskewness(X, Y + X, Y + 2*X) 16*sqrt(85)/85 >>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3) 9*sqrt(170)/85 >>> coskewness(Y, Y, Y) == skewness(Y) True >>> coskewness(X, Y + p*X, Y + 2*p*X) 4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2))) Returns ======= coskewness : The coskewness of the three random variables References ========== .. [1] https://en.wikipedia.org/wiki/Coskewness )rr%)r5rLZr8r9numZdenr:r:r; coskewnesss6 r`)rN)N)N)N)N)N)N)N)N)N)N)T)N);Z sympy.setsrZsympy.core.numbersrZsympy.core.relationalrZsympy.core.symbolrZ(sympy.functions.combinatorial.factorialsrZ&sympy.functions.elementary.exponentialrrZ(sympy.functions.elementary.miscellaneousr Z$sympy.functions.elementary.piecewiser Zsympy.integrals.integralsr Zsympy.solvers.solvesetr rvrrrrrrrrrrrrrrrrrrr __all__r-r$r?r%r)r(r+r.r/r&r'r,r*r`r!r"r#r:r:r:r;sf         T    * ' #  % - ' @ >