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| from sympy.concrete.summations import Sum | |
| from sympy.core.mul import Mul | |
| from sympy.core.numbers import (oo, pi) | |
| from sympy.core.relational import Eq | |
| from sympy.core.symbol import (Dummy, symbols) | |
| from sympy.functions.elementary.exponential import exp | |
| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.functions.elementary.trigonometric import sin | |
| from sympy.integrals.integrals import Integral | |
| from sympy.core.expr import unchanged | |
| from sympy.stats import (Normal, Poisson, variance, Covariance, Variance, | |
| Probability, Expectation, Moment, CentralMoment) | |
| from sympy.stats.rv import probability, expectation | |
| def test_literal_probability(): | |
| X = Normal('X', 2, 3) | |
| Y = Normal('Y', 3, 4) | |
| Z = Poisson('Z', 4) | |
| W = Poisson('W', 3) | |
| x = symbols('x', real=True) | |
| y, w, z = symbols('y, w, z') | |
| assert Probability(X > 0).evaluate_integral() == probability(X > 0) | |
| assert Probability(X > x).evaluate_integral() == probability(X > x) | |
| assert Probability(X > 0).rewrite(Integral).doit() == probability(X > 0) | |
| assert Probability(X > x).rewrite(Integral).doit() == probability(X > x) | |
| assert Expectation(X).evaluate_integral() == expectation(X) | |
| assert Expectation(X).rewrite(Integral).doit() == expectation(X) | |
| assert Expectation(X**2).evaluate_integral() == expectation(X**2) | |
| assert Expectation(x*X).args == (x*X,) | |
| assert Expectation(x*X).expand() == x*Expectation(X) | |
| assert Expectation(2*X + 3*Y + z*X*Y).expand() == 2*Expectation(X) + 3*Expectation(Y) + z*Expectation(X*Y) | |
| assert Expectation(2*X + 3*Y + z*X*Y).args == (2*X + 3*Y + z*X*Y,) | |
| assert Expectation(sin(X)) == Expectation(sin(X)).expand() | |
| assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y).expand() == 2*x*Expectation(sin(X)*Y) \ | |
| + y*Expectation(X**2) + z*Expectation(X*Y) | |
| assert Expectation(X + Y).expand() == Expectation(X) + Expectation(Y) | |
| assert Expectation((X + Y)*(X - Y)).expand() == Expectation(X**2) - Expectation(Y**2) | |
| assert Expectation((X + Y)*(X - Y)).expand().doit() == -12 | |
| assert Expectation(X + Y, evaluate=True).doit() == 5 | |
| assert Expectation(X + Expectation(Y)).doit() == 5 | |
| assert Expectation(X + Expectation(Y)).doit(deep=False) == 2 + Expectation(Expectation(Y)) | |
| assert Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False) == 2 \ | |
| + Expectation(Expectation(Y + Expectation(2*X))) | |
| assert Expectation(X + Expectation(Y + Expectation(2*X))).doit() == 9 | |
| assert Expectation(Expectation(2*X)).doit() == 4 | |
| assert Expectation(Expectation(2*X)).doit(deep=False) == Expectation(2*X) | |
| assert Expectation(4*Expectation(2*X)).doit(deep=False) == 4*Expectation(2*X) | |
| assert Expectation((X + Y)**3).expand() == 3*Expectation(X*Y**2) +\ | |
| 3*Expectation(X**2*Y) + Expectation(X**3) + Expectation(Y**3) | |
| assert Expectation((X - Y)**3).expand() == 3*Expectation(X*Y**2) -\ | |
| 3*Expectation(X**2*Y) + Expectation(X**3) - Expectation(Y**3) | |
| assert Expectation((X - Y)**2).expand() == -2*Expectation(X*Y) +\ | |
| Expectation(X**2) + Expectation(Y**2) | |
| assert Variance(w).args == (w,) | |
| assert Variance(w).expand() == 0 | |
| assert Variance(X).evaluate_integral() == Variance(X).rewrite(Integral).doit() == variance(X) | |
| assert Variance(X + z).args == (X + z,) | |
| assert Variance(X + z).expand() == Variance(X) | |
| assert Variance(X*Y).args == (Mul(X, Y),) | |
| assert type(Variance(X*Y)) == Variance | |
| assert Variance(z*X).expand() == z**2*Variance(X) | |
| assert Variance(X + Y).expand() == Variance(X) + Variance(Y) + 2*Covariance(X, Y) | |
| assert Variance(X + Y + Z + W).expand() == (Variance(X) + Variance(Y) + Variance(Z) + Variance(W) + | |
| 2 * Covariance(X, Y) + 2 * Covariance(X, Z) + 2 * Covariance(X, W) + | |
| 2 * Covariance(Y, Z) + 2 * Covariance(Y, W) + 2 * Covariance(W, Z)) | |
| assert Variance(X**2).evaluate_integral() == variance(X**2) | |
| assert unchanged(Variance, X**2) | |
| assert Variance(x*X**2).expand() == x**2*Variance(X**2) | |
| assert Variance(sin(X)).args == (sin(X),) | |
| assert Variance(sin(X)).expand() == Variance(sin(X)) | |
| assert Variance(x*sin(X)).expand() == x**2*Variance(sin(X)) | |
| assert Covariance(w, z).args == (w, z) | |
| assert Covariance(w, z).expand() == 0 | |
| assert Covariance(X, w).expand() == 0 | |
| assert Covariance(w, X).expand() == 0 | |
| assert Covariance(X, Y).args == (X, Y) | |
| assert type(Covariance(X, Y)) == Covariance | |
| assert Covariance(z*X + 3, Y).expand() == z*Covariance(X, Y) | |
| assert Covariance(X, X).args == (X, X) | |
| assert Covariance(X, X).expand() == Variance(X) | |
| assert Covariance(z*X + 3, w*Y + 4).expand() == w*z*Covariance(X,Y) | |
| assert Covariance(X, Y) == Covariance(Y, X) | |
| assert Covariance(X + Y, Z + W).expand() == Covariance(W, X) + Covariance(W, Y) + Covariance(X, Z) + Covariance(Y, Z) | |
| assert Covariance(x*X + y*Y, z*Z + w*W).expand() == (x*w*Covariance(W, X) + w*y*Covariance(W, Y) + | |
| x*z*Covariance(X, Z) + y*z*Covariance(Y, Z)) | |
| assert Covariance(x*X**2 + y*sin(Y), z*Y*Z**2 + w*W).expand() == (w*x*Covariance(W, X**2) + w*y*Covariance(sin(Y), W) + | |
| x*z*Covariance(Y*Z**2, X**2) + y*z*Covariance(Y*Z**2, sin(Y))) | |
| assert Covariance(X, X**2).expand() == Covariance(X, X**2) | |
| assert Covariance(X, sin(X)).expand() == Covariance(sin(X), X) | |
| assert Covariance(X**2, sin(X)*Y).expand() == Covariance(sin(X)*Y, X**2) | |
| assert Covariance(w, X).evaluate_integral() == 0 | |
| def test_probability_rewrite(): | |
| X = Normal('X', 2, 3) | |
| Y = Normal('Y', 3, 4) | |
| Z = Poisson('Z', 4) | |
| W = Poisson('W', 3) | |
| x, y, w, z = symbols('x, y, w, z') | |
| assert Variance(w).rewrite(Expectation) == 0 | |
| assert Variance(X).rewrite(Expectation) == Expectation(X ** 2) - Expectation(X) ** 2 | |
| assert Variance(X, condition=Y).rewrite(Expectation) == Expectation(X ** 2, Y) - Expectation(X, Y) ** 2 | |
| assert Variance(X, Y) != Expectation(X**2) - Expectation(X)**2 | |
| assert Variance(X + z).rewrite(Expectation) == Expectation((X + z) ** 2) - Expectation(X + z) ** 2 | |
| assert Variance(X * Y).rewrite(Expectation) == Expectation(X ** 2 * Y ** 2) - Expectation(X * Y) ** 2 | |
| assert Covariance(w, X).rewrite(Expectation) == -w*Expectation(X) + Expectation(w*X) | |
| assert Covariance(X, Y).rewrite(Expectation) == Expectation(X*Y) - Expectation(X)*Expectation(Y) | |
| assert Covariance(X, Y, condition=W).rewrite(Expectation) == Expectation(X * Y, W) - Expectation(X, W) * Expectation(Y, W) | |
| w, x, z = symbols("W, x, z") | |
| px = Probability(Eq(X, x)) | |
| pz = Probability(Eq(Z, z)) | |
| assert Expectation(X).rewrite(Probability) == Integral(x*px, (x, -oo, oo)) | |
| assert Expectation(Z).rewrite(Probability) == Sum(z*pz, (z, 0, oo)) | |
| assert Variance(X).rewrite(Probability) == Integral(x**2*px, (x, -oo, oo)) - Integral(x*px, (x, -oo, oo))**2 | |
| assert Variance(Z).rewrite(Probability) == Sum(z**2*pz, (z, 0, oo)) - Sum(z*pz, (z, 0, oo))**2 | |
| assert Covariance(w, X).rewrite(Probability) == \ | |
| -w*Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) + Integral(w*x*Probability(Eq(X, x)), (x, -oo, oo)) | |
| # To test rewrite as sum function | |
| assert Variance(X).rewrite(Sum) == Variance(X).rewrite(Integral) | |
| assert Expectation(X).rewrite(Sum) == Expectation(X).rewrite(Integral) | |
| assert Covariance(w, X).rewrite(Sum) == 0 | |
| assert Covariance(w, X).rewrite(Integral) == 0 | |
| assert Variance(X, condition=Y).rewrite(Probability) == Integral(x**2*Probability(Eq(X, x), Y), (x, -oo, oo)) - \ | |
| Integral(x*Probability(Eq(X, x), Y), (x, -oo, oo))**2 | |
| def test_symbolic_Moment(): | |
| mu = symbols('mu', real=True) | |
| sigma = symbols('sigma', positive=True) | |
| x = symbols('x') | |
| X = Normal('X', mu, sigma) | |
| M = Moment(X, 4, 2) | |
| assert M.rewrite(Expectation) == Expectation((X - 2)**4) | |
| assert M.rewrite(Probability) == Integral((x - 2)**4*Probability(Eq(X, x)), | |
| (x, -oo, oo)) | |
| k = Dummy('k') | |
| expri = Integral(sqrt(2)*(k - 2)**4*exp(-(k - \ | |
| mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)) | |
| assert M.rewrite(Integral).dummy_eq(expri) | |
| assert M.doit() == (mu**4 - 8*mu**3 + 6*mu**2*sigma**2 + \ | |
| 24*mu**2 - 24*mu*sigma**2 - 32*mu + 3*sigma**4 + 24*sigma**2 + 16) | |
| M = Moment(2, 5) | |
| assert M.doit() == 2**5 | |
| def test_symbolic_CentralMoment(): | |
| mu = symbols('mu', real=True) | |
| sigma = symbols('sigma', positive=True) | |
| x = symbols('x') | |
| X = Normal('X', mu, sigma) | |
| CM = CentralMoment(X, 6) | |
| assert CM.rewrite(Expectation) == Expectation((X - Expectation(X))**6) | |
| assert CM.rewrite(Probability) == Integral((x - Integral(x*Probability(True), | |
| (x, -oo, oo)))**6*Probability(Eq(X, x)), (x, -oo, oo)) | |
| k = Dummy('k') | |
| expri = Integral(sqrt(2)*(k - Integral(sqrt(2)*k*exp(-(k - \ | |
| mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)))**6*exp(-(k - \ | |
| mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)) | |
| assert CM.rewrite(Integral).dummy_eq(expri) | |
| assert CM.doit().simplify() == 15*sigma**6 | |
| CM = Moment(5, 5) | |
| assert CM.doit() == 5**5 | |