def test_single_normal():
mu = Symbol('mu', real=True)
sigma = Symbol('sigma', positive=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert E(Y) == mu
assert variance(Y) == sigma**2
pdf = density(Y)
x = Symbol('x', real=True)
assert (pdf(x) ==
2**S.Half*exp(-(x - mu)**2/(2*sigma**2))/(2*pi**S.Half*sigma))
assert P(X**2 < 1) == erf(2**S.Half/2)
assert quantile(Y)(x) == Intersection(S.Reals, FiniteSet(sqrt(2)*sigma*(sqrt(2)*mu/(2*sigma) + erfinv(2*x - 1))))
assert E(X, Eq(X, mu)) == mu
assert median(X) == FiniteSet(0)
# issue 8248
assert X.pspace.compute_expectation(1).doit() == 1
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
assert density(m)(1, 2) == 1/(2*pi)
raises (ValueError, lambda:m[2])
raises (ValueError,\
lambda: Normal('M',[1, 2], [[0, 0], [0, 1]]))
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N = Normal('N', [1, 2], [[x, 0], [0, y]])
assert density(N)(0, 0) == exp(-2/y - 1/(2*x))/(2*pi*sqrt(x*y))
raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
# symbolic
n = symbols('n', natural=True)
mu = MatrixSymbol('mu', n, 1)
sigma = MatrixSymbol('sigma', n, n)
X = Normal('X', mu, sigma)
assert density(X) == MultivariateNormalDistribution(mu, sigma)
raises (NotImplementedError, lambda: median(m))
def test_dagum():
p = Symbol("p", positive=True)
b = Symbol("b", positive=True)
a = Symbol("a", positive=True)
X = Dagum('x', p, a, b)
assert density(X)(x) == a*p*(x/b)**(a*p)*((x/b)**a + 1)**(-p - 1)/x
assert cdf(X)(x) == Piecewise(((1 + (x/b)**(-a))**(-p), x >= 0),
(0, True))
p = Symbol("p", nonpositive=True)
raises(ValueError, lambda: Dagum('x', p, a, b))
p = Symbol("p", positive=True)
b = Symbol("b", nonpositive=True)
raises(ValueError, lambda: Dagum('x', p, a, b))
b = Symbol("b", positive=True)
a = Symbol("a", nonpositive=True)
raises(ValueError, lambda: Dagum('x', p, a, b))
X = Dagum('x', 1 , 1, 1)
assert median(X) == FiniteSet(1)
def test_PowerFunction():
alpha = Symbol("alpha", nonpositive=True)
a, b = symbols('a, b', real=True)
raises (ValueError, lambda: PowerFunction('x', alpha, a, b))
a, b = symbols('a, b', real=False)
raises (ValueError, lambda: PowerFunction('x', alpha, a, b))
alpha = Symbol("alpha", positive=True)
a, b = symbols('a, b', real=True)
raises (ValueError, lambda: PowerFunction('x', alpha, 5, 2))
X = PowerFunction('X', 2, a, b)
assert density(X)(z) == (-2*a + 2*z)/(-a + b)**2
assert cdf(X)(z) == Piecewise((a**2/(a**2 - 2*a*b + b**2) -
2*a*z/(a**2 - 2*a*b + b**2) + z**2/(a**2 - 2*a*b + b**2), a <= z), (0, True))
X = PowerFunction('X', 2, 0, 1)
assert density(X)(z) == 2*z
assert cdf(X)(z) == Piecewise((z**2, z >= 0), (0,True))
assert E(X) == Rational(2,3)
assert P(X < 0) == 0
assert P(X < 1) == 1
assert median(X) == FiniteSet(1/sqrt(2))
def test_uniform():
l = Symbol('l', real=True)
w = Symbol('w', positive=True)
X = Uniform('x', l, l + w)
assert E(X) == l + w/2
assert variance(X).expand() == 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) == S.Half
assert median(X) == FiniteSet(4)
z = Symbol('z')
p = density(X)(z)
assert p.subs(z, 3.7) == S.Half
assert p.subs(z, -1) == 0
assert p.subs(z, 6) == 0
c = cdf(X)
assert c(2) == 0 and c(3) == 0
assert c(Rational(7, 2)) == Rational(1, 4)
assert c(5) == 1 and c(6) == 1
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
A = MultivariateNormal('A', [1, 2], [[1, 0], [0, 1]])
assert m == A
assert density(m)(1, 2) == 1 / (2 * pi)
assert m.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
raises(ValueError, lambda: m[2])
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1 / (2 * pi)
raises(ValueError, lambda: marginal_distribution(m))
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N = Normal('N', [1, 2], [[x, 0], [0, y]])
assert density(N)(0, 0) == exp(-((4 * x + y) /
(2 * x * y))) / (2 * pi * sqrt(x * y))
raises(ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
# symbolic
n = symbols('n', integer=True, positive=True)
mu = MatrixSymbol('mu', n, 1)
sigma = MatrixSymbol('sigma', n, n)
X = Normal('X', mu, sigma)
assert density(X) == MultivariateNormalDistribution(mu, sigma)
raises(NotImplementedError, lambda: median(m))
# Below tests should work after issue #17267 is resolved
# assert E(X) == mu
# assert variance(X) == sigma
# test symbolic multivariate normal densities
n = 3
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X = density(X)
eval_a = density_X(obs).subs({
Sg: eye(3),
mu: Matrix([0, 0, 0]),
obs: Matrix([0, 0, 0])
}).doit()
eval_b = density_X(0, 0, 0).subs({
Sg: eye(3),
mu: Matrix([0, 0, 0])
}).doit()
assert eval_a == sqrt(2) / (4 * pi**Rational(3 / 2))
assert eval_b == sqrt(2) / (4 * pi**Rational(3 / 2))
n = symbols('n', integer=True, positive=True)
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X_at_obs = density(X)(obs)
expected_density = MatrixElement(
exp((S(1)/2) * (mu.T - obs.T) * Sg**(-1) * (-mu + obs)) / \
sqrt((2*pi)**n * Determinant(Sg)), 0, 0)
assert density_X_at_obs == expected_density
def test_median_constant():
assert median(3) == 3
x = Symbol('x')
assert median(x) == x
def test_dice():
# TODO: Make iid method!
X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
a, b, t, p = symbols('a b t p')
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) is 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 + Y, 4) == kurtosis(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 quantile(X)(p) == Piecewise((nan, (p > 1) | (p < 0)),\
(S.One, p <= Rational(1, 6)), (S(2), p <= Rational(1, 3)), (S(3), p <= S.Half),\
(S(4), p <= Rational(2, 3)), (S(5), p <= Rational(5, 6)), (S(6), p <= 1))
assert P(X > 3, X > 3) is S.One
assert P(X > Y, Eq(Y, 6)) is S.Zero
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[S(22)] == Rational(1, 108) and d[S(4100)] == Rational(
1, 216) and S(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)
assert characteristic_function(X)(t) == exp(6 * I * t) / 6 + exp(
5 * I * t) / 6 + exp(4 * I * t) / 6 + exp(3 * I * t) / 6 + exp(
2 * I * t) / 6 + exp(I * t) / 6
assert moment_generating_function(X)(
t) == exp(6 * t) / 6 + exp(5 * t) / 6 + exp(4 * t) / 6 + exp(
3 * t) / 6 + exp(2 * t) / 6 + exp(t) / 6
assert median(X) == FiniteSet(3, 4)
D = Die('D', 7)
assert median(D) == FiniteSet(4)
# Bayes test for die
BayesTest(X > 3, X + Y < 5)
BayesTest(Eq(X - Y, Z), Z > Y)
BayesTest(X > 3, X > 2)
# arg test for die
raises(ValueError, lambda: Die('X', -1)) # issue 8105: negative sides.
raises(ValueError, lambda: Die('X', 0))
raises(ValueError, lambda: Die('X', 1.5)) # issue 8103: non integer sides.
# symbolic test for die
n, k = symbols('n, k', positive=True)
D = Die('D', n)
dens = density(D).dict
assert dens == Density(DieDistribution(n))
assert set(dens.subs(n, 4).doit().keys()) == {1, 2, 3, 4}
assert set(dens.subs(n, 4).doit().values()) == {Rational(1, 4)}
k = Dummy('k', integer=True)
assert E(D).dummy_eq(Sum(Piecewise((k / n, k <= n), (0, True)), (k, 1, n)))
assert variance(D).subs(n, 6).doit() == Rational(35, 12)
ki = Dummy('ki')
cumuf = cdf(D)(k)
assert cumuf.dummy_eq(
Sum(Piecewise((1 / n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, k)))
assert cumuf.subs({n: 6, k: 2}).doit() == Rational(1, 3)
t = Dummy('t')
cf = characteristic_function(D)(t)
assert cf.dummy_eq(
Sum(Piecewise((exp(ki * I * t) / n, (ki >= 1) & (ki <= n)), (0, True)),
(ki, 1, n)))
assert cf.subs(
n,
3).doit() == exp(3 * I * t) / 3 + exp(2 * I * t) / 3 + exp(I * t) / 3
mgf = moment_generating_function(D)(t)
assert mgf.dummy_eq(
Sum(Piecewise((exp(ki * t) / n, (ki >= 1) & (ki <= n)), (0, True)),
(ki, 1, n)))
assert mgf.subs(n,
3).doit() == exp(3 * t) / 3 + exp(2 * t) / 3 + exp(t) / 3
