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_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_verify_parameters():
pytest.raises(ValueError, lambda: Binomial('b', .2, .5))
pytest.raises(ValueError, lambda: Binomial('b', 3, 1.5))
def compute_moment(self, k):
if self.distribution == 'finite':
return sum([p * (b**k) for b, p in self.parameters])
if self.distribution == 'uniform':
l, u = self.parameters
return (u**(k + 1) - l**(k + 1)) / ((k + 1) * (u - l))
if self.distribution == 'gauss' or self.distribution == 'normal':
mu, sigma_squared = self.parameters
# For low moments avoid scipy.stats.moments as it does not support
# parametric parameters. In the future get all moments directly,
# using the following properties:
# https://math.stackexchange.com/questions/1945448/methods-for-finding-raw-moments-of-the-normal-distribution
if k == 0:
return 1
elif k == 1:
return mu
elif k == 2:
return mu**2 + sigma_squared
elif k == 3:
return mu * (mu**2 + 3 * sigma_squared)
elif k == 4:
return mu**4 + 6 * mu**2 * sigma_squared + 3 * sigma_squared**2
moment = norm(loc=mu, scale=sqrt(sigma_squared)).moment(k)
return Rational(moment)
if self.distribution == 'bernoulli':
return sympify(self.parameters[0])
if self.distribution == 'geometric':
p = sympify(self.parameters[0])
return p * polylog(-k, 1 - p)
if self.distribution == 'exponential':
lambd = sympify(self.parameters[0])
return factorial(k) / (lambd**k)
if self.distribution == 'beta':
alpha, beta = self.parameters
alpha = sympify(alpha)
beta = sympify(beta)
r = symbols('r')
return product((alpha + r) / (alpha + beta + r), (r, 0, k - 1))
if self.distribution == 'chi-squared':
n = sympify(self.parameters[0])
i = symbols('i')
return product(n + 2 * i, (i, 0, k - 1))
if self.distribution == 'rayleigh':
s = sympify(self.parameters[0])
return (2**(k / 2)) * (s**k) * gamma(1 + k / 2)
if self.distribution == 'unknown':
return sympify(f"{self.var_name}(0)^{k}")
if self.distribution == 'laplace':
mu, b = self.parameters
mu = sympify(mu)
b = sympify(b)
x = Laplace("x", mu, b)
return E(x**k)
if self.distribution == 'binomial':
n, p = self.parameters
n = sympify(n)
p = sympify(p)
x = Binomial("x", n, p)
return E(x**k)
if self.distribution == 'hypergeometric':
N, K, n = self.parameters
N = sympify(N)
K = sympify(K)
n = sympify(n)
x = Hypergeometric("x", N, K, n)
return E(x**k)