ó 9­\c@sddlZddlmZmZmZmZmZmZmZm Z ddl m Z ddl m Z ddlmZddlmZmZddlmZmZmZdd d d gZdefd „ƒYZd efd „ƒYZd efd„ƒYZd efd„ƒYZdS(iÿÿÿÿN(tExprtAddtMultStIntegraltEqtSumtSymbol(tdefault_sort_key(tglobal_evaluate(t_sympify(tvariancet covariance(t RandomSymbolt probabilityt expectationt Probabilityt ExpectationtVariancet CovariancecBs;eZdZdd„Zdd„Zdd„Zd„ZRS(s Symbolic expression for the probability. Examples ======== >>> from sympy.stats import Probability, Normal >>> from sympy import Integral >>> X = Normal("X", 0, 1) >>> prob = Probability(X > 1) >>> prob Probability(X > 1) Integral representation: >>> prob.rewrite(Integral) Integral(sqrt(2)*exp(-_z**2/2)/(2*sqrt(pi)), (_z, 1, oo)) Evaluation of the integral: >>> prob.evaluate_integral() sqrt(2)*(-sqrt(2)*sqrt(pi)*erf(sqrt(2)/2) + sqrt(2)*sqrt(pi))/(4*sqrt(pi)) cKs[t|ƒ}|dkr-tj||ƒ}n!t|ƒ}tj|||ƒ}||_|S(N(R tNoneRt__new__t _condition(tclstprobt conditiontkwargstobj((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR%s    cKst||dtƒS(Ntevaluate(RtFalse(tselftargRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyt_eval_rewrite_as_Integral/scKst||dtƒS(NR(RR(RRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyt_eval_rewrite_as_Sum2scCs|jtƒjƒS(N(trewriteRtdoit(R((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pytevaluate_integral5sN(t__name__t __module__t__doc__RRR R!R$(((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR s   cBsPeZdZdd„Zd„Zdd„Zdd„Zdd„Zd„Z RS(sÔ Symbolic expression for the expectation. Examples ======== >>> from sympy.stats import Expectation, Normal, Probability >>> from sympy import symbols, Integral >>> mu = symbols("mu") >>> sigma = symbols("sigma", positive=True) >>> X = Normal("X", mu, sigma) >>> Expectation(X) Expectation(X) >>> Expectation(X).evaluate_integral().simplify() mu To get the integral expression of the expectation: >>> Expectation(X).rewrite(Integral) Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) The same integral expression, in more abstract terms: >>> Expectation(X).rewrite(Probability) Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) This class is aware of some properties of the expectation: >>> from sympy.abc import a >>> Expectation(a*X) Expectation(a*X) >>> Y = Normal("Y", 0, 1) >>> Expectation(X + Y) Expectation(X + Y) To expand the ``Expectation`` into its expression, use ``doit()``: >>> Expectation(X + Y).doit() Expectation(X) + Expectation(Y) >>> Expectation(a*X + Y).doit() a*Expectation(X) + Expectation(Y) >>> Expectation(a*X + Y) Expectation(a*X + Y) cKsnt|ƒ}|dkr@|jtƒs+|Stj||ƒ}n!t|ƒ}tj|||ƒ}||_|S(N(R RthasR RRR(RtexprRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRgs    cKsø|jd}|j}|jtƒs)|St|tƒrjtg|jD]}t|d|ƒjƒ^qEŒSt|tƒrôg}g}xL|jD]A}t|tƒs³|jtƒrÃ|j |ƒq|j |ƒqWt|Œtt|Œd|ƒS|S(NiR( targsRR(R t isinstanceRRR#Rtappend(RthintsR)Rtatrvtnonrv((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR#ss  2 cKsv|jtƒ}t|ƒdkr-tƒ‚nt|ƒdkrC|S|jƒ}|jdkrmtdƒ‚n|j}|j dj ƒr¡t |j j ƒƒ}nt |j dƒ}|jj rt|j||ƒtt||ƒ|ƒ||jjjj|jjjjfƒS|jjr%t‚nMt|j||ƒtt||ƒ|ƒ||jjjj|jjjfƒSdS(NiisProbability space not knownt_1(tatomsR tlentNotImplementedErrortpoptpspaceRt ValueErrortsymboltnametisupperRtlowert is_ContinuousRtreplaceRRtdomaintsettinftsupt is_FiniteR(RRRRtrvsR/R8((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyt_eval_rewrite_as_Probabilityˆs"    P  cKst|d|dtƒS(NRR(RR(RRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR ¡scKs |jtƒS(N(R"R(RRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR!¤scCs|jtƒjƒS(N(R"RR#(R((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR$§sN( R%R&R'RRR#RDR R!R$(((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR9s,    cBs\eZdZdd„Zd„Zdd„Zdd„Zdd„Zdd„Z d„Z RS( s° Symbolic expression for the variance. Examples ======== >>> from sympy import symbols, Integral >>> from sympy.stats import Normal, Expectation, Variance, Probability >>> mu = symbols("mu", positive=True) >>> sigma = symbols("sigma", positive=True) >>> X = Normal("X", mu, sigma) >>> Variance(X) Variance(X) >>> Variance(X).evaluate_integral() sigma**2 Integral representation of the underlying calculations: >>> Variance(X).rewrite(Integral) Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**2*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)) Integral representation, without expanding the PDF: >>> Variance(X).rewrite(Probability) -Integral(x*Probability(Eq(X, x)), (x, -oo, oo))**2 + Integral(x**2*Probability(Eq(X, x)), (x, -oo, oo)) Rewrite the variance in terms of the expectation >>> Variance(X).rewrite(Expectation) -Expectation(X)**2 + Expectation(X**2) Some transformations based on the properties of the variance may happen: >>> from sympy.abc import a >>> Y = Normal("Y", 0, 1) >>> Variance(a*X) Variance(a*X) To expand the variance in its expression, use ``doit()``: >>> Variance(a*X).doit() a**2*Variance(X) >>> Variance(X + Y) Variance(X + Y) >>> Variance(X + Y).doit() 2*Covariance(X, Y) + Variance(X) + Variance(Y) cKs[t|ƒ}|dkr-tj||ƒ}n!t|ƒ}tj|||ƒ}||_|S(N(R RRRR(RRRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRÜs    c  sv|jd}|j‰|jtƒs,tjSt|tƒr?|St|tƒrÝg}x0|jD]%}|jtƒr^|j|ƒq^q^Wtt ‡fd†|ƒŒ}‡fd†}tt |t j |dƒƒŒ}||St|t ƒrrg}g}xA|jD]6}|jtƒr'|j|ƒq|j|dƒqWt |ƒdkrUtjSt |Œtt |ŒˆƒS|S(Nicst|ˆƒjƒS(N(RR#(txv(R(s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pytôtcsdtdˆ|ŒjƒS(NiR(RR#(tx(R(s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRFõRGi(R*RR(R RtZeroR+RR,tmapt itertoolst combinationsRR3R( RR-RR/R.t variancest map_to_covart covariancesR0((Rs?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR#æs4  !cKs.t|d|ƒ}t||ƒd}||S(Ni(R(RRRRte1te2((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyt_eval_rewrite_as_ExpectationscKs|jtƒjtƒS(N(R"RR(RRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRD scKst|jd|jdtƒS(NiR(R R*RR(RRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR scKs |jtƒS(N(R"R(RRRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR!scCs|jtƒjƒS(N(R"RR#(R((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR$sN( R%R&R'RRR#RRRDR R!R$(((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR«s0 !    cBszeZdZd d„Zd„Zed„ƒZed„ƒZd d„Z d d„Z d d„Z d d„Z d „Z RS( s Symbolic expression for the covariance. Examples ======== >>> from sympy.stats import Covariance >>> from sympy.stats import Normal >>> X = Normal("X", 3, 2) >>> Y = Normal("Y", 0, 1) >>> Z = Normal("Z", 0, 1) >>> W = Normal("W", 0, 1) >>> cexpr = Covariance(X, Y) >>> cexpr Covariance(X, Y) Evaluate the covariance, `X` and `Y` are independent, therefore zero is the result: >>> cexpr.evaluate_integral() 0 Rewrite the covariance expression in terms of expectations: >>> from sympy.stats import Expectation >>> cexpr.rewrite(Expectation) Expectation(X*Y) - Expectation(X)*Expectation(Y) In order to expand the argument, use ``doit()``: >>> from sympy.abc import a, b, c, d >>> Covariance(a*X + b*Y, c*Z + d*W) Covariance(a*X + b*Y, c*Z + d*W) >>> Covariance(a*X + b*Y, c*Z + d*W).doit() a*c*Covariance(X, Z) + a*d*Covariance(W, X) + b*c*Covariance(Y, Z) + b*d*Covariance(W, Y) This class is aware of some properties of the covariance: >>> Covariance(X, X).doit() Variance(X) >>> Covariance(a*X, b*Y).doit() a*b*Covariance(X, Y) cKs¤t|ƒ}t|ƒ}|jdtdƒrOt||gdtƒ\}}n|dkrstj|||ƒ}n$t|ƒ}tj||||ƒ}||_|S(NRitkey( R R5R tsortedRRRRR(Rtarg1targ2RRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRFs  !   c KsF|jd}|jd}|j}||krBt||ƒjƒS|jtƒsXtjS|jtƒsntjSt||gdt ƒ\}}t |tƒrºt |tƒrºt |||ƒS|j |j ƒƒ}|j |j ƒƒ}g|D]K\}}|D]8\} } || t d|t|| gdt ƒŒ^qûqë} t| ŒS(NiiRSR(R*RRR#R(R RRIRTRR+Rt_expand_single_argumenttexpandR( RR-RURVRtcoeff_rv_list1tcoeff_rv_list2R.tr1tbtr2taddends((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR#Us"    UcCsÞt|tƒrtj|fgSt|tƒrœg}xa|jD]V}t|tƒrl|j|j|ƒƒq>t|tƒr>|jtj|fƒq>q>W|St|tƒr»|j|ƒgS|j tƒrÚtj|fgSdS(N( R+R RtOneRR*RR,t_get_mul_nonrv_rv_tupleR((RR)toutvalR.((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRWnscCsbg}g}x=|jD]2}|jtƒr;|j|ƒq|j|ƒqWt|Œt|ŒfS(N(R*R(R R,R(RtmR/R0R.((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR`scKs7t|||ƒ}t||ƒt||ƒ}||S(N(R(RRURVRRRPRQ((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRRŒscKs|jtƒjtƒS(N(R"RR(RRURVRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRD‘scKs't|jd|jd|jdtƒS(NiiR(R R*RR(RRURVRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR ”scKs |jtƒS(N(R"R(RRURVRR((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR!—scCs|jtƒjƒS(N(R"RR#(R((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyR$šsN(R%R&R'RRR#t classmethodRWR`RRRDR R!R$(((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyRs+      (RKtsympyRRRRRRRRtsympy.core.compatibilityRtsympy.core.evaluateR tsympy.core.sympifyR t sympy.statsR R tsympy.stats.rvR RRt__all__RRRR(((s?lib/python2.7/site-packages/sympy/stats/symbolic_probability.pyts :,rn