ó ¡¼™\c@ sKddlmZmZddlmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZddlmZddddd d d d d dddddddddddddgZdd$d„Zd$d„Zd$d„ZeZd$d„Zd$d „Zd$d!„Zd$d"„Zd$d#„ZeZ eZ!d$S(%iÿÿÿÿ(tprint_functiontdivisioni(t probabilityt expectationtdensitytwheretgiventpspacetcdftcharacteristic_functiontsamplet sample_itertrandom_symbolst independentt dependenttsampling_densitytmoment_generating_function(tsqrttPtERRRR RR RR tvariancetstdtskewnesst covarianceRR R t correlationtmomenttcmomentRRicK st|||||S(s4 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(tXtntct conditiontkwargs((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyRscK st|d||S(sn 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*(1 - p) i(R(RRR((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyR#scK stt|||ƒS(sG 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*(1 - p)) (RR(RRR((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pytstandard_deviation<scK s4t|t||||t|||||S(s" 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(RtYRR((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyRRscK s3t||||t|||t|||S(s˜ 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!RR((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyRqs!cK s(t|||}t|||||S(s< 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(RRRRtmu((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyRscK s1t|||}d||t||||S(sÊ 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 i(RR(RRRRtsigma((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pytsmoment¥scK st|d||S(sý 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 i(R$(RRR((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyR»sN("t __future__RRtrvRRRRRRRR R R R R RRRtsympyRt__all__tNoneRRR RRRRR$RRR(((s7lib/python2.7/site-packages/sympy/stats/rv_interface.pyts"d