def prove(Eq):
assert NonnegativeIntegers.is_extended_negative == False
assert NonnegativeIntegers.is_extended_nonnegative
lamda0 = Symbol.lamda0(positive=True)
lamda1 = Symbol.lamda1(positive=True)
x0 = Symbol.x0(distribution=PoissonDistribution(lamda0))
x1 = Symbol.x1(distribution=PoissonDistribution(lamda1))
Eq << apply(x0, x1)
Eq << Eq[0].lhs.this.doit(evaluate=False)
Eq << Eq[-1].subs(Eq[0])
Eq << Eq[-1].this.rhs.powsimp()
y = Eq[0].lhs.symbol
Eq << Eq[-1] * factorial(y)
Eq << axiom.discrete.combinatorics.binomial.theorem.apply(
lamda0, lamda1, y)
Eq << Eq[-1].subs(Eq[-2])
Eq << Eq[-1].this.rhs.combsimp()
def test_Compound_Distribution():
X = Normal('X', 2, 4)
N = NormalDistribution(X, 4)
C = CompoundDistribution(N)
assert C.is_Continuous
assert C.set == Interval(-oo, oo)
assert C.pdf(x, evaluate=True).simplify() == exp(-x**2/64 + x/16 - S(1)/16)/(8*sqrt(pi))
assert not isinstance(CompoundDistribution(NormalDistribution(2, 3)),
CompoundDistribution)
M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]])
raises(NotImplementedError, lambda: CompoundDistribution(M))
X = Beta('X', 2, 4)
B = BernoulliDistribution(X, 1, 0)
C = CompoundDistribution(B)
assert C.is_Finite
assert C.set == {0, 1}
y = symbols('y', negative=False, integer=True)
assert C.pdf(y, evaluate=True) == Piecewise((S(1)/(30*beta(2, 4)), Eq(y, 0)),
(S(1)/(60*beta(2, 4)), Eq(y, 1)), (0, True))
k, t, z = symbols('k t z', positive=True, real=True)
G = Gamma('G', k, t)
X = PoissonDistribution(G)
C = CompoundDistribution(X)
assert C.is_Discrete
assert C.set == S.Naturals0
assert C.pdf(z, evaluate=True).simplify() == t**z*(t + 1)**(-k - z)*gamma(k \
+ z)/(gamma(k)*gamma(z + 1))
def test_density():
x = Symbol('x')
l = Symbol('l', positive=True)
rate = Beta(l, 2, 3)
X = Poisson(x, rate)
assert isinstance(pspace(X), ProductPSpace)
assert density(X, Eq(rate, rate.symbol)) == PoissonDistribution(l)
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2 * x, evaluate=False), Sum)
def test_PoissonDistribution():
l = 3
p = PoissonDistribution(l)
assert abs(p.cdf(10).evalf() - 1) < .001
assert abs(p.cdf(10.4).evalf() - 1) < .001
assert p.expectation(x, x) == l
assert p.expectation(x**2, x) - p.expectation(x, x)**2 == l
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2 * x, evaluate=False), Sum)
assert characteristic_function(x)(0).doit() == 1
def test_Poisson():
l = 3
x = Poisson("x", l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Sum)
assert isinstance(E(2 * x, evaluate=False), Sum)
# issue 8248
assert x.pspace.compute_expectation(1) == 1
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
assert isinstance(E(x, evaluate=False), Expectation)
assert isinstance(E(2*x, evaluate=False), Expectation)
# issue 8248
assert x.pspace.compute_expectation(1) == 1
def test_density():
x = Symbol('x')
l = Symbol('l', positive=True)
rate = Beta(l, 2, 3)
X = Poisson(x, rate)
assert isinstance(pspace(X), JointPSpace)
assert density(X, Eq(rate, rate.symbol)) == PoissonDistribution(l)
N1 = Normal('N1', 0, 1)
N2 = Normal('N2', N1, 2)
assert density(N2)(0).doit() == sqrt(10) / (10 * sqrt(pi))
assert simplify(density(N2, Eq(N1, 1))(x)) == \
sqrt(2)*exp(-(x - 1)**2/8)/(4*sqrt(pi))
def apply(x0, x1):
assert x0.is_random and x1.is_random
pspace0 = pspace(x0)
pspace1 = pspace(x1)
if not isinstance(pspace0, SingleDiscretePSpace) or not isinstance(
pspace1, SingleDiscretePSpace):
return None
distribution0 = pspace0.distribution
distribution1 = pspace1.distribution
if not isinstance(distribution0, PoissonDistribution) or not isinstance(
distribution1, PoissonDistribution):
return None
Y = Symbol.y(distribution=PoissonDistribution(distribution0.lamda +
distribution1.lamda))
y = pspace(Y).symbol
return Equality(PDF(x0 + x1)(y), PDF(Y)(y).doit())
def test_compound_pspace():
X = Normal('X', 2, 4)
Y = Normal('Y', 3, 6)
assert not isinstance(Y.pspace, CompoundPSpace)
N = NormalDistribution(1, 2)
D = PoissonDistribution(3)
B = BernoulliDistribution(0.2, 1, 0)
pspace1 = CompoundPSpace('N', N)
pspace2 = CompoundPSpace('D', D)
pspace3 = CompoundPSpace('B', B)
assert not isinstance(pspace1, CompoundPSpace)
assert not isinstance(pspace2, CompoundPSpace)
assert not isinstance(pspace3, CompoundPSpace)
M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]])
raises(ValueError, lambda: CompoundPSpace('M', M))
Y = Normal('Y', X, 6)
assert isinstance(Y.pspace, CompoundPSpace)
assert Y.pspace.distribution == CompoundDistribution(NormalDistribution(X, 6))
assert Y.pspace.domain.set == Interval(-oo, oo)
def test_PoissonProcess():
X = PoissonProcess("X", 3)
assert X.state_space == S.Naturals0
assert X.index_set == Interval(0, oo)
assert X.lamda == 3
t, d, x, y = symbols('t d x y', positive=True)
assert isinstance(X(t), RandomIndexedSymbol)
assert X.distribution(t) == PoissonDistribution(3 * t)
raises(ValueError, lambda: PoissonProcess("X", -1))
raises(NotImplementedError, lambda: X[t])
raises(IndexError, lambda: X(-5))
assert X.joint_distribution(X(2), X(3)) == JointDistributionHandmade(
Lambda((X(2), X(3)), 6**X(2) * 9**X(3) * exp(-15) /
(factorial(X(2)) * factorial(X(3)))))
assert X.joint_distribution(4, 6) == JointDistributionHandmade(
Lambda((X(4), X(6)), 12**X(4) * 18**X(6) * exp(-30) /
(factorial(X(4)) * factorial(X(6)))))
assert P(X(t) < 1) == exp(-3 * t)
assert P(Eq(X(t), 0),
Contains(t, Interval.Lopen(3, 5))) == exp(-6) # exp(-2*lamda)
res = P(Eq(X(t), 1), Contains(t, Interval.Lopen(3, 4)))
assert res == 3 * exp(-3)
# Equivalent to P(Eq(X(t), 1))**4 because of non-overlapping intervals
assert P(
Eq(X(t), 1) & Eq(X(d), 1) & Eq(X(x), 1) & Eq(X(y), 1),
Contains(t, Interval.Lopen(0, 1))
& Contains(d, Interval.Lopen(1, 2)) & Contains(x, Interval.Lopen(2, 3))
& Contains(y, Interval.Lopen(3, 4))) == res**4
# Return Probability because of overlapping intervals
assert P(Eq(X(t), 2) & Eq(X(d), 3), Contains(t, Interval.Lopen(0, 2))
& Contains(d, Interval.Ropen(2, 4))) == \
Probability(Eq(X(d), 3) & Eq(X(t), 2), Contains(t, Interval.Lopen(0, 2))
& Contains(d, Interval.Ropen(2, 4)))
raises(ValueError, lambda: P(
Eq(X(t), 2) & Eq(X(d), 3),
Contains(t, Interval.Lopen(0, 4)) & Contains(d, Interval.Lopen(3, oo)))
) # no bound on d
assert P(Eq(X(3), 2)) == 81 * exp(-9) / 2
assert P(Eq(X(t), 2), Contains(t, Interval.Lopen(0,
5))) == 225 * exp(-15) / 2
# Check that probability works correctly by adding it to 1
res1 = P(X(t) <= 3, Contains(t, Interval.Lopen(0, 5)))
res2 = P(X(t) > 3, Contains(t, Interval.Lopen(0, 5)))
assert res1 == 691 * exp(-15)
assert (res1 + res2).simplify() == 1
# Check Not and Or
assert P(Not(Eq(X(t), 2) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & \
Contains(d, Interval.Lopen(7, 8))).simplify() == -18*exp(-6) + 234*exp(-9) + 1
assert P(Eq(X(t), 2) | Ne(X(t), 4),
Contains(t, Interval.Ropen(2, 4))) == 1 - 36 * exp(-6)
raises(ValueError, lambda: P(X(t) > 2, X(t) + X(d)))
assert E(
X(t)) == 3 * t # property of the distribution at a given timestamp
assert E(
X(t)**2 + X(d) * 2 + X(y)**3,
Contains(t, Interval.Lopen(0, 1))
& Contains(d, Interval.Lopen(1, 2))
& Contains(y, Interval.Ropen(3, 4))) == 75
assert E(X(t)**2, Contains(t, Interval.Lopen(0, 1))) == 12
assert E(x*(X(t) + X(d))*(X(t)**2+X(d)**2), Contains(t, Interval.Lopen(0, 1))
& Contains(d, Interval.Ropen(1, 2))) == \
Expectation(x*(X(d) + X(t))*(X(d)**2 + X(t)**2), Contains(t, Interval.Lopen(0, 1))
& Contains(d, Interval.Ropen(1, 2)))
# Value Error because of infinite time bound
raises(ValueError, lambda: E(X(t)**3, Contains(t, Interval.Lopen(1, oo))))
# Equivalent to E(X(t)**2) - E(X(d)**2) == E(X(1)**2) - E(X(1)**2) == 0
assert E((X(t) + X(d)) * (X(t) - X(d)),
Contains(t, Interval.Lopen(0, 1))
& Contains(d, Interval.Lopen(1, 2))) == 0
assert E(X(2) + x * E(X(5))) == 15 * x + 6
assert E(x * X(1) + y) == 3 * x + y
assert P(Eq(X(1), 2) & Eq(X(t), 3),
Contains(t, Interval.Lopen(1, 2))) == 81 * exp(-6) / 4
Y = PoissonProcess("Y", 6)
Z = X + Y
assert Z.lamda == X.lamda + Y.lamda == 9
raises(ValueError,
lambda: X + 5) # should be added be only PoissonProcess instance
N, M = Z.split(4, 5)
assert N.lamda == 4
assert M.lamda == 5
raises(ValueError, lambda: Z.split(3, 2)) # 2+3 != 9
raises(
ValueError, lambda: P(Eq(X(t), 0),
Contains(t, Interval.Lopen(1, 3)) & Eq(X(1), 0)))
# check if it handles queries with two random variables in one args
res1 = P(Eq(N(3), N(5)))
assert res1 == P(Eq(N(t), 0), Contains(t, Interval(3, 5)))
res2 = P(N(3) > N(1))
assert res2 == P((N(t) > 0), Contains(t, Interval(1, 3)))
assert P(N(3) < N(1)) == 0 # condition is not possible
res3 = P(N(3) <= N(1)) # holds only for Eq(N(3), N(1))
assert res3 == P(Eq(N(t), 0), Contains(t, Interval(1, 3)))
# tests from https://www.probabilitycourse.com/chapter11/11_1_2_basic_concepts_of_the_poisson_process.php
X = PoissonProcess('X', 10) # 11.1
assert P(Eq(X(S(1) / 3), 3)
& Eq(X(1), 10)) == exp(-10) * Rational(8000000000, 11160261)
assert P(Eq(X(1), 1), Eq(X(S(1) / 3), 3)) == 0
assert P(Eq(X(1), 10), Eq(X(S(1) / 3), 3)) == P(Eq(X(S(2) / 3), 7))
X = PoissonProcess('X', 2) # 11.2
assert P(X(S(1) / 2) < 1) == exp(-1)
assert P(X(3) < 1, Eq(X(1), 0)) == exp(-4)
assert P(Eq(X(4), 3), Eq(X(2), 3)) == exp(-4)
X = PoissonProcess('X', 3)
assert P(Eq(X(2), 5) & Eq(X(1), 2)) == Rational(81, 4) * exp(-6)
# check few properties
assert P(
X(2) <= 3,
X(1) >= 1) == 3 * P(Eq(X(1), 0)) + 2 * P(Eq(X(1), 1)) + P(Eq(X(1), 2))
assert P(X(2) <= 3, X(1) > 1) == 2 * P(Eq(X(1), 0)) + 1 * P(Eq(X(1), 1))
assert P(Eq(X(2), 5) & Eq(X(1), 2)) == P(Eq(X(1), 3)) * P(Eq(X(1), 2))
assert P(Eq(X(3), 4), Eq(X(1), 3)) == P(Eq(X(2), 1))
#test issue 20078
assert (2 * X(t) + 3 * X(t)).simplify() == 5 * X(t)
assert (2 * X(t) - 3 * X(t)).simplify() == -X(t)
assert (2 * (0.25 * X(t))).simplify() == 0.5 * X(t)
assert (2 * X(t) * 0.25 * X(t)).simplify() == 0.5 * X(t)**2
assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1) * X(t)**2
def test_PoissonDistribution():
l = 3
p = PoissonDistribution(l)
assert abs(p.cdf(10).evalf() - 1) < .001
assert p.expectation(x, x) == l
assert p.expectation(x**2, x) - p.expectation(x, x)**2 == l
