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_prefab_sampling():
N = Normal('X', 0, 1)
L = LogNormal('L', 0, 1)
E = Exponential('Ex', 1)
P = Pareto('P', 1, 3)
W = Weibull('W', 1, 1)
U = Uniform('U', 0, 1)
B = Beta('B', 2, 5)
G = Gamma('G', 1, 3)
variables = [N, L, E, P, W, U, B, G]
niter = 10
for var in variables:
for i in range(niter):
assert sample(var) in var.pspace.domain.set
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)
# issue sympy/sympy#10003
X = Exponential('x', 3)
G = Gamma('g', 1, 2)
assert P(X < -1) == 0
assert P(G < -1) == 0
def test_sympyissue_10003():
X = Exponential('x', 3)
G = Gamma('g', 1, 2)
assert P(X < -1) == 0
assert P(G < -1) == 0
def test_issue_10003():
X = Exponential('x', 3)
G = Gamma('g', 1, 2)
assert P(X < -1) == S.Zero
assert P(G < -1) == S.Zero