def test_discreteuniform():
# Symbolic
a, b, c = symbols('a b c')
X = DiscreteUniform('X', [a, b, c])
assert X.pspace.distribution.pdf(a) == Rational(1, 3)
assert X.pspace.distribution.pdf(p) == 0
assert E(X) == (a + b + c)/3
assert simplify(variance(X)
- ((a**2 + b**2 + c**2)/3 - (a/3 + b/3 + c/3)**2)) == 0
assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == Rational(1, 3)
Y = DiscreteUniform('Y', range(-5, 5))
# Numeric
assert E(Y) == -Rational(1, 2)
assert variance(Y) == Rational(33, 4)
for x in range(-5, 5):
assert P(Eq(Y, x)) == Rational(1, 10)
assert P(Y <= x) == Rational(x + 6, 10)
assert P(Y >= x) == Rational(5 - x, 10)
assert dict(density(Die('D', 6)).items()) == \
dict(density(DiscreteUniform('U', range(1, 7))).items())
def test_discreteuniform():
# Symbolic
a, b, c = symbols('a b c')
X = DiscreteUniform('X', [a, b, c])
assert X.pspace.distribution.pdf(a) == Rational(1, 3)
assert X.pspace.distribution.pdf(p) == 0
assert E(X) == (a + b + c) / 3
assert simplify(
variance(X) - ((a**2 + b**2 + c**2) / 3 -
(a / 3 + b / 3 + c / 3)**2)) == 0
assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == Rational(1, 3)
Y = DiscreteUniform('Y', range(-5, 5))
# Numeric
assert E(Y) == -Rational(1, 2)
assert variance(Y) == Rational(33, 4)
for x in range(-5, 5):
assert P(Eq(Y, x)) == Rational(1, 10)
assert P(Y <= x) == Rational(x + 6, 10)
assert P(Y >= x) == Rational(5 - x, 10)
assert dict(density(Die('D', 6)).items()) == \
dict(density(DiscreteUniform('U', range(1, 7))).items())
def test_ContinuousRV():
x = Symbol('x')
pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
# X and Y should be equivalent
X = ContinuousRV(x, pdf)
Y = Normal('y', 0, 1)
assert variance(X) == variance(Y)
assert P(X > 0) == P(Y > 0)
def test_ContinuousRV():
x = Symbol('x')
pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
# X and Y should be equivalent
X = ContinuousRV(x, pdf)
Y = Normal('y', 0, 1)
assert variance(X) == variance(Y)
assert P(X > 0) == P(Y > 0)
def test_bernoulli():
p, a, b = symbols('p a b')
X = Bernoulli('B', p, a, b)
assert E(X) == a*p + b*(-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
X = Bernoulli('B', p, 1, 0)
assert E(X) == p
assert simplify(variance(X)) == p*(1 - p)
assert E(a*X + b) == a*E(X) + b
assert simplify(variance(a*X + b)) == simplify(a**2 * variance(X))
def test_bernoulli():
p, a, b = symbols('p a b')
X = Bernoulli('B', p, a, b)
assert E(X) == a * p + b * (-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
X = Bernoulli('B', p, 1, 0)
assert E(X) == p
assert simplify(variance(X)) == p * (1 - p)
assert E(a * X + b) == a * E(X) + b
assert simplify(variance(a * X + b)) == simplify(a**2 * variance(X))
def test_pareto_numeric():
xm, beta = Integer(3), Integer(2)
alpha = beta + 5
X = Pareto('x', xm, alpha)
assert E(X) == alpha*xm/(alpha - 1)
assert variance(X) == xm**2*alpha/(((alpha - 1)**2*(alpha - 2)))
def test_rademacher():
X = Rademacher('X')
assert E(X) == 0
assert variance(X) == 1
assert density(X)[-1] == Rational(1, 2)
assert density(X)[1] == Rational(1, 2)
def test_rademacher():
X = Rademacher('X')
assert E(X) == 0
assert variance(X) == 1
assert density(X)[-1] == S.Half
assert density(X)[1] == S.Half
def test_rayleigh():
sigma = Symbol("sigma", positive=True)
X = Rayleigh('x', sigma)
assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
def test_rademacher():
X = Rademacher('X')
assert E(X) == 0
assert variance(X) == 1
assert density(X)[-1] == Rational(1, 2)
assert density(X)[1] == Rational(1, 2)
def test_pareto_numeric():
xm, beta = Integer(3), Integer(2)
alpha = beta + 5
X = Pareto('x', xm, alpha)
assert E(X) == alpha*xm/(alpha - 1)
assert variance(X) == xm**2*alpha/(((alpha - 1)**2*(alpha - 2)))
def test_rayleigh():
sigma = Symbol('sigma', positive=True)
X = Rayleigh('x', sigma)
assert density(X)(x) == x*exp(-x**2/(2*sigma**2))/sigma**2
assert E(X) == sqrt(2)*sqrt(pi)*sigma/2
assert variance(X) == -pi*sigma**2/2 + 2*sigma**2
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_maxwell():
a = Symbol("a", positive=True)
X = Maxwell('x', a)
assert density(X)(x) == (sqrt(2)*x**2*exp(-x**2/(2*a**2)) /
(sqrt(pi)*a**3))
assert E(X) == 2*sqrt(2)*a/sqrt(pi)
assert simplify(variance(X)) == a**2*(-8 + 3*pi)/pi
def test_weibull_numeric():
# Test for integers and rationals
a = 1
bvals = [Rational(1, 2), 1, Rational(3, 2), Integer(5)]
for b in bvals:
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(
a**2 * gamma(1 + 2/b) - E(X)**2)
def test_symbolic():
mu1, mu2 = symbols('mu1 mu2', real=True)
s1, s2 = symbols('sigma1 sigma2', real=True, positive=True)
rate = Symbol('lambda', real=True, positive=True)
X = Normal('x', mu1, s1)
Y = Normal('y', mu2, s2)
Z = Exponential('z', rate)
a, b, c = symbols('a b c', real=True)
assert E(X) == mu1
assert E(X + Y) == mu1 + mu2
assert E(a*X + b) == a*E(X) + b
assert variance(X) == s1**2
assert simplify(variance(X + a*Y + b)) == variance(X) + a**2*variance(Y)
assert E(Z) == 1/rate
assert E(a*Z + b) == a*E(Z) + b
assert E(X + a*Z + b) == mu1 + a/rate + b
def test_symbolic():
mu1, mu2 = symbols('mu1 mu2', real=True)
s1, s2 = symbols('sigma1 sigma2', real=True, positive=True)
rate = Symbol('lambda', real=True, positive=True)
X = Normal('x', mu1, s1)
Y = Normal('y', mu2, s2)
Z = Exponential('z', rate)
a, b, c = symbols('a b c', real=True)
assert E(X) == mu1
assert E(X + Y) == mu1 + mu2
assert E(a*X + b) == a*E(X) + b
assert variance(X) == s1**2
assert simplify(variance(X + a*Y + b)) == variance(X) + a**2*variance(Y)
assert E(Z) == 1/rate
assert E(a*Z + b) == a*E(Z) + b
assert E(X + a*Z + b) == mu1 + a/rate + b
def test_weibull_numeric():
# Test for integers and rationals
a = 1
bvals = [Rational(1, 2), 1, Rational(3, 2), Integer(5)]
for b in bvals:
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(
a**2 * gamma(1 + 2/b) - E(X)**2)
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 density(x)(z) == 3**z/(exp(3)*factorial(z))
def test_maxwell():
a = Symbol('a', positive=True)
X = Maxwell('x', a)
assert density(X)(x) == (sqrt(2)*x**2*exp(-x**2/(2*a**2)) /
(sqrt(pi)*a**3))
assert E(X) == 2*sqrt(2)*a/sqrt(pi)
assert simplify(variance(X)) == a**2*(-8 + 3*pi)/pi
def test_dice():
# TODO: Make iid method!
X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
a, b = symbols('a b')
assert E(X) == 3 + Rational(1, 2)
assert variance(X) == Rational(35, 12)
assert E(X + Y) == 7
assert E(X + X) == 7
assert E(a * X + b) == a * E(X) + b
assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2)
assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2)
assert cmoment(X, 0) == 1
assert cmoment(4 * X, 3) == 64 * cmoment(X, 3)
assert covariance(X, Y) == 0
assert covariance(X, X + Y) == variance(X)
assert density(Eq(cos(X * pi), 1))[True] == Rational(1, 2)
assert correlation(X, Y) == 0
assert correlation(X, Y) == correlation(Y, X)
assert smoment(X + Y, 3) == skewness(X + Y)
assert smoment(X, 0) == 1
assert P(X > 3) == Rational(1, 2)
assert P(2 * X > 6) == Rational(1, 2)
assert P(X > Y) == Rational(5, 12)
assert P(Eq(X, Y)) == P(Eq(X, 1))
assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3)
assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3)
assert E(X + Y, Eq(X, Y)) == E(2 * X)
assert moment(X, 0) == 1
assert moment(5 * X, 2) == 25 * moment(X, 2)
assert P(X > 3, X > 3) == 1
assert P(X > Y, Eq(Y, 6)) == 0
assert P(Eq(X + Y, 12)) == Rational(1, 36)
assert P(Eq(X + Y, 12), Eq(X, 6)) == Rational(1, 6)
assert density(X + Y) == density(Y + Z) != density(X + X)
d = density(2 * X + Y**Z)
assert d[22] == Rational(1, 108) and d[4100] == Rational(
1, 216) and 3130 not in d
assert pspace(X).domain.as_boolean() == Or(
*[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]])
assert where(X > 3).set == FiniteSet(4, 5, 6)
X = Die('X', 2)
x = X.symbol
assert X.pspace.compute_cdf(X) == {1: Rational(1, 2), 2: 1}
assert X.pspace.sorted_cdf(X) == [(1, Rational(1, 2)), (2, 1)]
assert X.pspace.compute_density(X)(1) == Rational(1, 2)
assert X.pspace.compute_density(X)(0) == 0
assert X.pspace.compute_density(X)(8) == 0
assert X.pspace.density == x
def test_dice():
# TODO: Make iid method!
X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
a, b = symbols('a b')
assert E(X) == 3 + Rational(1, 2)
assert variance(X) == Rational(35, 12)
assert E(X + Y) == 7
assert E(X + X) == 7
assert E(a*X + b) == a*E(X) + b
assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2)
assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2)
assert cmoment(X, 0) == 1
assert cmoment(4*X, 3) == 64*cmoment(X, 3)
assert covariance(X, Y) == 0
assert covariance(X, X + Y) == variance(X)
assert density(Eq(cos(X*pi), 1))[True] == Rational(1, 2)
assert correlation(X, Y) == 0
assert correlation(X, Y) == correlation(Y, X)
assert smoment(X + Y, 3) == skewness(X + Y)
assert smoment(X, 0) == 1
assert P(X > 3) == Rational(1, 2)
assert P(2*X > 6) == Rational(1, 2)
assert P(X > Y) == Rational(5, 12)
assert P(Eq(X, Y)) == P(Eq(X, 1))
assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3)
assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3)
assert E(X + Y, Eq(X, Y)) == E(2*X)
assert moment(X, 0) == 1
assert moment(5*X, 2) == 25*moment(X, 2)
assert P(X > 3, X > 3) == 1
assert P(X > Y, Eq(Y, 6)) == 0
assert P(Eq(X + Y, 12)) == Rational(1, 36)
assert P(Eq(X + Y, 12), Eq(X, 6)) == Rational(1, 6)
assert density(X + Y) == density(Y + Z) != density(X + X)
d = density(2*X + Y**Z)
assert d[22] == Rational(1, 108) and d[4100] == Rational(1, 216) and 3130 not in d
assert pspace(X).domain.as_boolean() == Or(
*[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]])
assert where(X > 3).set == FiniteSet(4, 5, 6)
X = Die('X', 2)
x = X.symbol
assert X.pspace.compute_cdf(X) == {1: Rational(1, 2), 2: 1}
assert X.pspace.sorted_cdf(X) == [(1, Rational(1, 2)), (2, 1)]
assert X.pspace.compute_density(X)(1) == Rational(1, 2)
assert X.pspace.compute_density(X)(0) == 0
assert X.pspace.compute_density(X)(8) == 0
assert X.pspace.density == x
def test_nakagami():
mu = Symbol("mu", positive=True)
omega = Symbol("omega", positive=True)
X = Nakagami('x', mu, omega)
assert density(X)(x) == (2*x**(2*mu - 1)*mu**mu*omega**(-mu)
* exp(-x**2*mu/omega)/gamma(mu))
assert simplify(E(X, meijerg=True)) == (sqrt(mu)*sqrt(omega)
* gamma(mu + Rational(1, 2))/gamma(mu + 1))
assert simplify(variance(X, meijerg=True)) == (
omega - omega*gamma(mu + Rational(1, 2))**2/(gamma(mu)*gamma(mu + 1)))
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == Rational(1, 2)
def test_nakagami():
mu = Symbol('mu', positive=True)
omega = Symbol('omega', positive=True)
X = Nakagami('x', mu, omega)
assert density(X)(x) == (2*x**(2*mu - 1)*mu**mu*omega**(-mu)
* exp(-x**2*mu/omega)/gamma(mu))
assert simplify(E(X, meijerg=True)) == (sqrt(mu)*sqrt(omega)
* gamma(mu + Rational(1, 2))/gamma(mu + 1))
assert simplify(variance(X, meijerg=True)) == (
omega - omega*gamma(mu + Rational(1, 2))**2/(gamma(mu)*gamma(mu + 1)))
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == Rational(1, 2)
def test_uniform():
l = Symbol('l', real=True)
w = Symbol('w', positive=True, finite=True)
X = Uniform('x', l, l + w)
assert simplify(E(X)) == l + w/2
assert simplify(variance(X)) == w**2/12
# With numbers all is well
X = Uniform('x', 3, 5)
assert P(X < 3) == 0 and P(X > 5) == 0
assert P(X < 4) == P(X > 4) == Rational(1, 2)
def test_binomial_symbolic():
n = 10 # Because we're using for loops, can't do symbolic n
p = symbols('p', positive=True)
X = Binomial('X', n, p)
assert simplify(E(X)) == n * p == simplify(moment(X, 1))
assert simplify(variance(X)) == n * p * (1 - p) == simplify(cmoment(X, 2))
assert cancel((skewness(X) - (1 - 2 * p) / sqrt(n * p * (1 - p)))) == 0
# Test ability to change success/failure winnings
H, T = symbols('H T')
Y = Binomial('Y', n, p, succ=H, fail=T)
assert simplify(E(Y) - (n * (H * p + T * (1 - p)))) == 0
def test_binomial_symbolic():
n = 10 # Because we're using for loops, can't do symbolic n
p = symbols('p', positive=True)
X = Binomial('X', n, p)
assert simplify(E(X)) == n*p == simplify(moment(X, 1))
assert simplify(variance(X)) == n*p*(1 - p) == simplify(cmoment(X, 2))
assert cancel((skewness(X) - (1-2*p)/sqrt(n*p*(1-p)))) == 0
# Test ability to change success/failure winnings
H, T = symbols('H T')
Y = Binomial('Y', n, p, succ=H, fail=T)
assert simplify(E(Y) - (n*(H*p + T*(1 - p)))) == 0
def test_uniform():
l = Symbol('l', real=True)
w = Symbol('w', positive=True, finite=True)
X = Uniform('x', l, l + w)
assert simplify(E(X)) == l + w/2
assert simplify(variance(X)) == w**2/12
# With numbers all is well
X = Uniform('x', 3, 5)
assert P(X < 3) == 0 and P(X > 5) == 0
assert P(X < 4) == P(X > 4) == Rational(1, 2)
def test_binomial_numeric():
nvals = range(5)
pvals = [0, Rational(1, 4), Rational(1, 2), Rational(3, 4), 1]
for n in nvals:
for p in pvals:
X = Binomial('X', n, p)
assert E(X) == n*p
assert variance(X) == n*p*(1 - p)
if n > 0 and 0 < p < 1:
assert skewness(X) == (1 - 2*p)/sqrt(n*p*(1 - p))
for k in range(n + 1):
assert P(Eq(X, k)) == binomial(n, k)*p**k*(1 - p)**(n - k)
def test_binomial_numeric():
nvals = range(5)
pvals = [0, Rational(1, 4), Rational(1, 2), Rational(3, 4), 1]
for n in nvals:
for p in pvals:
X = Binomial('X', n, p)
assert E(X) == n * p
assert variance(X) == n * p * (1 - p)
if n > 0 and 0 < p < 1:
assert skewness(X) == (1 - 2 * p) / sqrt(n * p * (1 - p))
for k in range(n + 1):
assert P(Eq(X, k)) == binomial(n, k) * p**k * (1 - p)**(n - k)
def test_hypergeometric_numeric():
for N in range(1, 5):
for m in range(N + 1):
for n in range(1, N + 1):
X = Hypergeometric('X', N, m, n)
N, m, n = map(sympify, (N, m, n))
assert sum(density(X).values()) == 1
assert E(X) == n * m / N
if N > 1:
assert variance(X) == n*(m/N)*(N - m)/N*(N - n)/(N - 1)
# Only test for skewness when defined
if N > 2 and 0 < m < N and n < N:
assert skewness(X) == simplify((N - 2*m)*sqrt(N - 1)*(N - 2*n)
/ (sqrt(n*m*(N - m)*(N - n))*(N - 2)))
def test_hypergeometric_numeric():
for N in range(1, 5):
for m in range(N + 1):
for n in range(1, N + 1):
X = Hypergeometric('X', N, m, n)
N, m, n = map(sympify, (N, m, n))
assert sum(density(X).values()) == 1
assert E(X) == n * m / N
if N > 1:
assert variance(X) == n*(m/N)*(N - m)/N*(N - n)/(N - 1)
# Only test for skewness when defined
if N > 2 and 0 < m < N and n < N:
assert skewness(X) == simplify((N - 2*m)*sqrt(N - 1)*(N - 2*n)
/ (sqrt(n*m*(N - m)*(N - n))*(N - 2)))
def test_exponential():
rate = Symbol('lambda', positive=True, real=True)
X = Exponential('x', rate)
assert E(X) == 1/rate
assert variance(X) == 1/rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert smoment(2*X, 4) == smoment(X, 4)
assert moment(X, 3) == 3*2*1/rate**3
assert P(X > 0) == Integer(1)
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10*rate)
assert where(X <= 1).set == Interval(0, 1)
def test_exponential():
rate = Symbol('lambda', positive=True, real=True)
X = Exponential('x', rate)
assert E(X) == 1/rate
assert variance(X) == 1/rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert smoment(2*X, 4) == smoment(X, 4)
assert moment(X, 3) == 3*2*1/rate**3
assert P(X > 0) == Integer(1)
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10*rate)
assert where(X <= 1).set == Interval(0, 1)
def test_single_normal():
mu = Symbol('mu', real=True)
sigma = Symbol('sigma', real=True, positive=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert simplify(E(Y)) == mu
assert simplify(variance(Y)) == sigma**2
pdf = density(Y)
x = Symbol('x')
assert (pdf(x) ==
sqrt(2)*exp(-(x - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma))
assert P(X**2 < 1) == erf(sqrt(2)/2)
assert E(X, Eq(X, mu)) == mu
def test_single_normal():
mu = Symbol('mu', real=True)
sigma = Symbol('sigma', real=True, positive=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert simplify(E(Y)) == mu
assert simplify(variance(Y)) == sigma**2
pdf = density(Y)
x = Symbol('x')
assert (pdf(x) ==
sqrt(2)*exp(-(x - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma))
assert P(X**2 < 1) == erf(sqrt(2)/2)
assert E(X, Eq(X, mu)) == mu
def test_gamma():
k = Symbol('k', positive=True)
theta = Symbol('theta', positive=True)
X = Gamma('x', k, theta)
assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k)
assert cdf(X, meijerg=True)(z) == Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) +
k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0),
(0, True))
# assert simplify(variance(X)) == k*theta**2 # handled numerically below
assert E(X) == moment(X, 1)
k, theta = symbols('k theta', real=True, positive=True)
X = Gamma('x', k, theta)
assert simplify(E(X)) == k*theta
# can't get things to simplify on this one so we use subs
assert variance(X).subs({k: 5}) == (k*theta**2).subs({k: 5})
def test_gamma():
k = Symbol("k", positive=True)
theta = Symbol("theta", positive=True)
X = Gamma('x', k, theta)
assert density(X)(x) == x**(k - 1)*theta**(-k)*exp(-x/theta)/gamma(k)
assert cdf(X, meijerg=True)(z) == Piecewise(
(-k*lowergamma(k, 0)/gamma(k + 1) +
k*lowergamma(k, z/theta)/gamma(k + 1), z >= 0),
(0, True))
# assert simplify(variance(X)) == k*theta**2 # handled numerically below
assert E(X) == moment(X, 1)
k, theta = symbols('k theta', real=True, positive=True)
X = Gamma('x', k, theta)
assert simplify(E(X)) == k*theta
# can't get things to simplify on this one so we use subs
assert variance(X).subs({k: 5}) == (k*theta**2).subs({k: 5})
def test_dice():
# TODO: Make iid method!
X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
a, b = symbols('a b')
assert E(X) == 3 + S.Half
assert variance(X) == Rational(35, 12)
assert E(X + Y) == 7
assert E(X + X) == 7
assert E(a * X + b) == a * E(X) + b
assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2)
assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2)
assert cmoment(X, 0) == 1
assert cmoment(4 * X, 3) == 64 * cmoment(X, 3)
assert covariance(X, Y) == S.Zero
assert covariance(X, X + Y) == variance(X)
assert density(Eq(cos(X * S.Pi), 1))[True] == S.Half
assert correlation(X, Y) == 0
assert correlation(X, Y) == correlation(Y, X)
assert smoment(X + Y, 3) == skewness(X + Y)
assert smoment(X, 0) == 1
assert P(X > 3) == S.Half
assert P(2 * X > 6) == S.Half
assert P(X > Y) == Rational(5, 12)
assert P(Eq(X, Y)) == P(Eq(X, 1))
assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3)
assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3)
assert E(X + Y, Eq(X, Y)) == E(2 * X)
assert moment(X, 0) == 1
assert moment(5 * X, 2) == 25 * moment(X, 2)
assert P(X > 3, X > 3) == S.One
assert P(X > Y, Eq(Y, 6)) == S.Zero
assert P(Eq(X + Y, 12)) == S.One / 36
assert P(Eq(X + Y, 12), Eq(X, 6)) == S.One / 6
assert density(X + Y) == density(Y + Z) != density(X + X)
d = density(2 * X + Y**Z)
assert d[Integer(22)] == S.One / 108 and d[Integer(
4100)] == S.One / 216 and Integer(3130) not in d
assert pspace(X).domain.as_boolean() == Or(
*[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]])
assert where(X > 3).set == FiniteSet(4, 5, 6)
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
# This is too slow
# assert E(B) == a / (a + b)
# assert variance(B) == (a*b) / ((a+b)**2 * (a+b+1))
# Full symbolic solution is too much, test with numeric version
a, b = Integer(1), Integer(2)
B = Beta('x', a, b)
assert expand_func(E(B)) == a/(a + b)
assert expand_func(variance(B)) == (a*b)/(a + b)**2/(a + b + 1)
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
# This is too slow
# assert E(B) == a / (a + b)
# assert variance(B) == (a*b) / ((a+b)**2 * (a+b+1))
# Full symbolic solution is too much, test with numeric version
a, b = Integer(1), Integer(2)
B = Beta('x', a, b)
assert expand_func(E(B)) == a/(a + b)
assert expand_func(variance(B)) == (a*b)/(a + b)**2/(a + b + 1)
def test_Sample():
X = Die('X', 6)
Y = Normal('Y', 0, 1)
z = Symbol('z')
assert sample(X) in [1, 2, 3, 4, 5, 6]
assert sample(X + Y).is_Float
P(X + Y > 0, Y < 0, numsamples=10).is_number
assert E(X + Y, numsamples=10).is_number
assert variance(X + Y, numsamples=10).is_number
pytest.raises(ValueError, lambda: P(Y > z, numsamples=5))
assert P(sin(Y) <= 1, numsamples=10) == 1
assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10) == 1
# Make sure this doesn't raise an error
E(Sum(1 / z**Y, (z, 1, oo)), Y > 2, numsamples=3)
assert all(i in range(1, 7) for i in density(X, numsamples=10))
assert all(i in range(4, 7) for i in density(X, X > 3, numsamples=10))
def test_Sample():
X = Die('X', 6)
Y = Normal('Y', 0, 1)
assert sample(X) in [1, 2, 3, 4, 5, 6]
assert sample(X + Y).is_Float
assert P(X + Y > 0, Y < 0, numsamples=10).is_number
assert P(X > 10, numsamples=10).is_number
assert E(X + Y, numsamples=10).is_number
assert variance(X + Y, numsamples=10).is_number
pytest.raises(TypeError, lambda: P(Y > z, numsamples=5))
assert P(sin(Y) <= 1, numsamples=10, modules=["math"]) == 1
assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10, modules=["math"]) == 1
assert all(i in range(1, 7) for i in density(X, numsamples=10))
assert all(i in range(4, 7) for i in density(X, X > 3, numsamples=10))
# Make sure this doesn't raise an error
Y = Normal('Y', 0, 1)
E(Sum(1/z**Y, (z, 1, oo)), Y > 2, numsamples=3, modules="mpmath")
def test_weibull():
a, b = symbols('a b', positive=True, real=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
def test_weibull():
a, b = symbols('a b', positive=True)
X = Weibull('x', a, b)
assert simplify(E(X)) == simplify(a * gamma(1 + 1/b))
assert simplify(variance(X)) == simplify(a**2 * gamma(1 + 2/b) - E(X)**2)
