+ io6^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t H t ^RI Ht^RIHt^R IHt^R IHt^R IHtHtHtHtHtHtHtHtHtHtH t H!t!H"t"H#t#H$t$H%t%H&t&H't'H(t(.ROt)RR R/Rllt*RRlt+RRlt,],t-RRlt.RRlt/RRlt0RR R/Rllt1RRlt2RRlt3RRlt4RRlt5RRlt6RRlt7]t8]t9].t:R #) ) 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_processNevaluateTc ^RIHpV'dV!WW#4P4#V!WW#4P\4#)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 )Moment) sympy.stats.symbolic_probabilityr#doitrewriter )Xnc conditionr!kwargsr#s&&&&$, x/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/stats/rv_interface.pymomentr-s8*8aA)..00 ! % - -h 77c \V4'd'\V4\48Xd^RIHpV!W4#\ V^V3/VB#)aE Variance of a random expression. .. math:: variance(X) = E((X-E(X))^{2}) Examples ======== >>> from sympy.stats import Die, Bernoulli, variance >>> from sympy import simplify, Symbol >>> X = Die('X', 6) >>> p = Symbol('p') >>> B = Bernoulli('B', p, 1, 0) >>> variance(2*X) 35/3 >>> simplify(variance(B)) p*(1 - p) )Variance)rrrr$r0cmoment)r'r*r+r0s&&, r,variancer25s<.||q VX-=%% 1a -f --r.c ,\\W3/VB4#)a 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 r2r'r*r+s&&,r,standard_deviationr5Ss& 00 11r.c a\W3/VBpVPR\^44o\V\4'd$\ V3RlVP 444#\\V!V4S4)4#)a  Calculates 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 bc3J<"TFq)\VS4,xK R#5iN)r ).0probbases& r, entropy..sFuSt_,,s #) rgetr isinstancedictsumvaluesrr )exprr*r+pdfr<s&&, @r,entropyrFisfH $ ,V ,C ::c3q6 "D#tFFF F CIt,, --r.c 0\V4'd\V4\48Xg*\V4'd(\V4\48Xd^RIHpV!WV4#\ V\ W3/VB, V\ W3/VB, ,V3/VB#)a 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 ) Covariance)rrrr$rHr)r'Yr*r+rHs&&&, r, covariancerJs: ! fh.IaLLVAYRXRZEZ?! **  [ 0 0 0 [ 0 0 0 2  r.c `\WV3/VB\W3/VB\W3/VB,, #)aa 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)) )rJstd)r'rIr*r+s&&&,r, correlationrMs;> aI 0 0#a2Mf2M 1"6"3# $$r.c ^RIHpV'dV!WV4P4#V!WV4P\4#)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 ) CentralMoment)r$rOr%r&r )r'r(r*r!r+rOs&&&$, r,r1r1s8&?Q9-2244 y ) 1 1( ;;r.c ^\W3/VBp^V, V,\WV3/VB,#)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 )rLr1)r'r(r*r+sigmas&&&, r,smomentrRs2*  ' 'E eGa<i:6: ::r.c  \V^3RV/VB#)a 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 r*rRr4s&&,r,skewnessrUsF 1a 79 7 77r.c  \V^3RV/VB#)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 r*rTr4s&&,r,kurtosisrW3sT 1a 79 7 77r.c 0\\W43RV/VB#)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 r*)rr)r'r(r*r+s&&&,r,factorial_momentrY`s J '- M Mf MMr.c \V4'gV#^RIHp^RIHp^RIHp\\V4V4'd\V4PV4p.pVP!4FkwrV \^^48gK^V , \V4P\W44,\^^48gKZVPV4Km \V!#\\V4W434'dh\V4PV4p\!R4p \#\%V!V 4\^^4, 4V \V4P&4pV#\)R\+\V44,4h)a Calculates 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 )ContinuousPSpace)DiscretePSpace) FinitePSpacexz$The median of %s is not implemented.)rsympy.stats.crvr[sympy.stats.drvr\sympy.stats.frvr]r@r compute_cdfitemsrrrappendrrr r setNotImplementedErrorstr) r'r!r+r[r\r]rresultkeyvaluer^s &&, r,medianrks*R Q<<0.,&)\**Qi##A&))+JCx1~%1u9 1I ! !"Q* -+.19!Q+@ c"&&!!&).?@@Qi##A& #J.Q(1a.)@A1fQimmT DSPQ^S TTr.c  \V\W3/VB, V\W3/VB, ,V\W#3/VB, ,V3/VBp\W3/VB\W3/VB,\W#3/VB,pWV, #)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 )rrL)r'rIZr*r+numdens&&&&, r, coskewnessrpsl q;q>v>> A3F3 35 A3F3 356? KCI KC a %f %A(CF(C C q &v & 'C 7Nr.)PEHrrrrrrrrr2rLrUrWrJrrFrkrrrMrYr-r1rrrRrr )rNr9)T); sympy.setsrsympy.core.numbersrsympy.core.relationalrsympy.core.symbolr(sympy.functions.combinatorial.factorialsr&sympy.functions.elementary.exponentialrr (sympy.functions.elementary.miscellaneousr $sympy.functions.elementary.piecewiser sympy.integrals.integralsr sympy.solvers.solvesetr rvrrrrrrrrrrrrrrrrrrr __all__r-r2r5rLrFrJrMr1rRrUrWrYrkrprqrrrsr.r,rs '$#E=9?.+,,,,,,  <8$86.<2((.T$N $F