B [@sddlmZmZddlmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZddlmZddddd d d d d dddddddddddddgZd&ddZd'ddZd(ddZeZd)d dZd*d!dZd+d"dZd,d#d$Zd-d%dZeZeZ dS).)print_functiondivision) probability expectationdensitywheregivenpspacecdfcharacteristic_functionsample sample_iterrandom_symbols independent dependentsampling_densitymoment_generating_function)sqrtPErrr r r r r rvariancestdskewness covariancerrr correlationmomentcmomentrrNcKst||||f|S)a4 Return the nth moment of a random expression about c i.e. E((X-c)**n) Default value of c is 0. Examples ======== >>> from sympy.stats import Die, moment, E >>> X = Die('X', 6) >>> moment(X, 1, 6) -5/2 >>> moment(X, 2) 91/6 >>> moment(X, 1) == E(X) True )r)Xnc conditionkwargsr#7lib/python3.7/site-packages/sympy/stats/rv_interface.pyrscKst|d|f|S)ao Variance of a random expression Expectation of (X-E(X))**2 Examples ======== >>> from sympy.stats import Die, E, Bernoulli, variance >>> from sympy import simplify, Symbol >>> X = Die('X', 6) >>> p = Symbol('p') >>> B = Bernoulli('B', p, 1, 0) >>> variance(2*X) 35/3 >>> simplify(variance(B)) p*(-p + 1) )r)rr!r"r#r#r$r#scKstt||f|S)aH Standard Deviation of a random expression Square root of the Expectation of (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*(-p + 1)) )rr)rr!r"r#r#r$standard_deviation<sr&cKs.t|t||f||t||f||f|S)a" Covariance of two random expressions The expectation that the two variables will rise and fall together Covariance(X,Y) = E( (X-E(X)) * (Y-E(Y)) ) Examples ======== >>> from sympy.stats import Exponential, covariance >>> from sympy import Symbol >>> rate = Symbol('lambda', positive=True, real=True, finite=True) >>> X = Exponential('X', rate) >>> Y = Exponential('Y', rate) >>> covariance(X, X) lambda**(-2) >>> covariance(X, Y) 0 >>> covariance(X, Y + rate*X) 1/lambda )r)rYr!r"r#r#r$rRscKs,t|||f|t||f|t||f|S)a Correlation of two random expressions, also known as correlation coefficient or Pearson's correlation The normalized expectation that the two variables will rise and fall together 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)) )rr)rr'r!r"r#r#r$rqscKs t||f|}t||||f|S)a< Return the nth central moment of a random expression about its mean i.e. E((X - E(X))**n) Examples ======== >>> from sympy.stats import Die, cmoment, variance >>> X = Die('X', 6) >>> cmoment(X, 3) 0 >>> cmoment(X, 2) 35/12 >>> cmoment(X, 2) == variance(X) True )rr)rrr!r"Zmur#r#r$rscKs*t||f|}d||t|||f|S)a Return the nth Standardized moment of a random expression i.e. 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)rr)rrr!r"Zsigmar#r#r$smomentsr(cKst|d|f|S)a Measure of the asymmetry of the probability distribution Positive skew indicates that most of the values lie to the right of the mean skewness(X) = E( ((X - E(X))/sigma)**3 ) Examples ======== >>> from sympy.stats import skewness, Exponential, Normal >>> from sympy import Symbol >>> X = Normal('X', 0, 1) >>> skewness(X) 0 >>> rate = Symbol('lambda', positive=True, real=True, finite=True) >>> Y = Exponential('Y', rate) >>> skewness(Y) 2 )r()rr!r"r#r#r$rs)rN)N)N)N)N)N)N)N)!Z __future__rrrvrrrrr r r r r rrrrrrZsympyr__all__rrr&rrrrr(rrrr#r#r#r$s"D