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Current File : //usr/local/lib/python3.6/dist-packages/sympy/stats/tests/test_symbolic_probability.py from sympy import symbols, Mul, sin, Integral, oo, Eq, Sum, sqrt, exp, pi, Dummy
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', real=True, 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
def test_symbolic_CentralMoment():
mu = symbols('mu', real=True)
sigma = symbols('sigma', real=True, 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
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