def test_literal_probability():
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 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).doit() == x*Expectation(X)
assert Expectation(2*X + 3*Y + z*X*Y).doit() == 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)).doit()
assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y).doit() == 2*x*Expectation(sin(X)*Y) + y*Expectation(X**2) + z*Expectation(X*Y)
assert Variance(w).args == (w,)
assert Variance(w).doit() == 0
assert Variance(X).evaluate_integral() == Variance(X).rewrite(Integral).doit() == variance(X)
assert Variance(X + z).args == (X + z,)
assert Variance(X + z).doit() == Variance(X)
assert Variance(X*Y).args == (Mul(X, Y),)
assert type(Variance(X*Y)) == Variance
assert Variance(z*X).doit() == z**2*Variance(X)
assert Variance(X + Y).doit() == Variance(X) + Variance(Y) + 2*Covariance(X, Y)
assert Variance(X + Y + Z + W).doit() == (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 Variance(X**2) == Variance(X**2)
assert Variance(x*X**2).doit() == x**2*Variance(X**2)
assert Variance(sin(X)).args == (sin(X),)
assert Variance(sin(X)).doit() == Variance(sin(X))
assert Variance(x*sin(X)).doit() == x**2*Variance(sin(X))
assert Covariance(w, z).args == (w, z)
assert Covariance(w, z).doit() == 0
assert Covariance(X, w).doit() == 0
assert Covariance(w, X).doit() == 0
assert Covariance(X, Y).args == (X, Y)
assert type(Covariance(X, Y)) == Covariance
assert Covariance(z*X + 3, Y).doit() == z*Covariance(X, Y)
assert Covariance(X, X).args == (X, X)
assert Covariance(X, X).doit() == Variance(X)
assert Covariance(z*X + 3, w*Y + 4).doit() == w*z*Covariance(X,Y)
assert Covariance(X, Y) == Covariance(Y, X)
assert Covariance(X + Y, Z + W).doit() == Covariance(W, X) + Covariance(W, Y) + Covariance(X, Z) + Covariance(Y, Z)
assert Covariance(x*X + y*Y, z*Z + w*W).doit() == (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).doit() == (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).doit() == Covariance(X, X**2)
assert Covariance(X, sin(X)).doit() == Covariance(sin(X), X)
assert Covariance(X**2, sin(X)*Y).doit() == Covariance(sin(X)*Y, X**2)
def test_literal_probability():
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 Probability(X > 0).doit() == probability(X > 0)
assert Probability(X > x).doit() == probability(X > x)
assert Expectation(X).doit() == expectation(X)
assert Expectation(X**2).doit() == expectation(X**2)
assert Expectation(x*X) == x*Expectation(X)
assert Expectation(2*X + 3*Y + z*X*Y) == 2*Expectation(X) + 3*Expectation(Y) + z*Expectation(X*Y)
assert Expectation(2*X + 3*Y + z*X*Y, evaluate=False).args == (2*X + 3*Y + z*X*Y,)
assert Expectation(sin(X)) == Expectation(sin(X), evaluate=False)
assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y) == 2*x*Expectation(sin(X)*Y) + y*Expectation(X**2) + z*Expectation(X*Y)
assert Variance(w) == 0
assert Variance(X).doit() == variance(X)
assert Variance(X + z) == Variance(X)
assert Variance(X*Y).args == (Mul(X, Y),)
assert type(Variance(X*Y)) == Variance
assert Variance(z*X) == z**2*Variance(X)
assert Variance(X + Y) == Variance(X) + Variance(Y) + 2*Covariance(X, Y)
assert Variance(X + Y + Z + W) == (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).doit() == variance(X**2)
assert Variance(X**2, evaluate=False) == Variance(X**2)
assert Variance(x*X**2) == x**2*Variance(X**2)
assert Variance(sin(X)).args == (sin(X),)
assert Variance(sin(X), evaluate=False) == Variance(sin(X))
assert Variance(x*sin(X)) == x**2*Variance(sin(X))
assert Covariance(w, z) == 0
assert Covariance(X, w) == 0
assert Covariance(w, X) == 0
assert Covariance(X, Y).args == (X, Y)
assert type(Covariance(X, Y)) == Covariance
assert Covariance(z*X + 3, Y) == z*Covariance(X, Y)
assert Covariance(X, X) == Variance(X)
assert Covariance(z*X + 3, w*Y + 4) == w*z*Covariance(X,Y)
assert Covariance(X, Y) == Covariance(Y, X)
assert Covariance(X + Y, Z + W) == Covariance(W, X) + Covariance(W, Y) + Covariance(X, Z) + Covariance(Y, Z)
assert Covariance(x*X + y*Y, z*Z + w*W) == (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) == (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) == Covariance(X, X**2, evaluate=False)
assert Covariance(X, sin(X)) == Covariance(sin(X), X, evaluate=False)
assert Covariance(X**2, sin(X)*Y) == Covariance(sin(X)*Y, X**2, evaluate=False)
def doit(self, **kwargs):
return expectation(self.args[0], condition=self._condition, **kwargs)
def _eval_rewrite_as_Integral(self, arg, condition=None):
return expectation(arg, condition=condition, evaluate=False)
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 _eval_rewrite_as_Integral(self, arg, condition=None):
return expectation(arg, condition=condition, evaluate=False)