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_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_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_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_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_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_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)
