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_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_lognormal():
mean = Symbol('mu', real=True)
std = Symbol('sigma', positive=True, real=True)
X = LogNormal('x', mean, std)
# The diofant integrator can't do this too well
# assert E(X) == exp(mean+std**2/2)
# assert variance(X) == (exp(std**2)-1) * exp(2*mean + std**2)
# Right now, only density function and sampling works
# Test sampling: Only e^mean in sample std of 0
for i in range(3):
X = LogNormal('x', i, 0)
assert sample(X) == N(exp(i))
# The diofant integrator can't do this too well
# assert E(X) ==
mu = Symbol("mu", extended_real=True)
sigma = Symbol("sigma", positive=True)
X = LogNormal('x', mu, sigma)
assert density(X)(x) == (sqrt(2)*exp(-(-mu + log(x))**2
/ (2*sigma**2))/(2*x*sqrt(pi)*sigma))
X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
assert density(X)(x) == sqrt(2)*exp(-log(x)**2/2)/(2*x*sqrt(pi))
def test_conditional_1d():
X = Normal('x', 0, 1)
Y = given(X, X >= 0)
assert density(Y)(x) == 2 * density(X)(x)
assert Y.pspace.domain.set == Interval(0, oo, right_open=True)
assert E(Y) == sqrt(2/pi)
assert E(X**2) == E(Y**2)
def test_difficult_univariate():
""" Since using solve in place of deltaintegrate we're able to perform
substantially more complex density computations on single continuous random
variables
"""
x = Normal('x', 0, 1)
assert density(x**3)
assert density(exp(x**2))
assert density(log(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_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_Density():
X = Die('X', 6)
d = Density(X)
assert d.doit() == density(X)
r = Rational(1, 6)
assert density(2*X).dict == {2: r, 4: r, 6: r, 8: r, 10: r, 12: r}
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 1)
assert str(density(X - Y, Y)(z)) == 'sqrt(2)*E**(-(Y + z)**2/2)/(2*sqrt(pi))'
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_frechet():
a = Symbol("a", positive=True)
s = Symbol("s", positive=True)
m = Symbol("m", extended_real=True)
X = Frechet("x", a, s=s, m=m)
assert density(X)(x) == a*((x - m)/s)**(-a - 1)*exp(-((x - m)/s)**(-a))/s
def test_f_distribution():
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=True)
X = FDistribution("x", d1, d2)
assert density(X)(x) == (d2**(d2/2)*sqrt((d1*x)**d1*(d1*x + d2)**(-d1 - d2))
/ (x*beta(d1/2, d2/2)))
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
def test_chi_noncentral():
k = Symbol("k", integer=True)
l = Symbol("l")
X = ChiNoncentral("x", k, l)
assert density(X)(x) == (x**k*l*(x*l)**(-k/2) *
exp(-x**2/2 - l**2/2)*besseli(k/2 - 1, x*l))
def test_coins():
C, D = Coin('C'), Coin('D')
H, T = symbols('H, T')
assert P(Eq(C, D)) == Rational(1, 2)
assert density(Tuple(C, D)) == {(H, H): Rational(1, 4), (H, T): Rational(1, 4),
(T, H): Rational(1, 4), (T, T): Rational(1, 4)}
assert dict(density(C).items()) == {H: Rational(1, 2), T: Rational(1, 2)}
F = Coin('F', Rational(1, 10))
assert P(Eq(F, H)) == Rational(1, 10)
d = pspace(C).domain
assert d.as_boolean() == Or(Eq(C.symbol, H), Eq(C.symbol, T))
pytest.raises(ValueError, lambda: P(C > D)) # Can't intelligently compare H to T
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_fisher_z():
d1 = Symbol("d1", positive=True)
d2 = Symbol("d2", positive=True)
X = FisherZ("x", d1, d2)
assert density(X)(x) == (2*d1**(d1/2)*d2**(d2/2) *
(d1*exp(2*x) + d2)**(-d1/2 - d2/2) *
exp(d1*x)/beta(d1/2, d2/2))
def test_quadratic_u():
a = Symbol("a", extended_real=True)
b = Symbol("b", extended_real=True)
X = QuadraticU("x", a, b)
assert density(X)(x) == (Piecewise((12*(x - a/2 - b/2)**2/(-a + b)**3,
And(x <= b, a <= x)), (0, True)))
assert X.pspace.domain.set == Interval(a, b)
def test_pareto():
xm, beta = symbols('xm beta', positive=True, finite=True)
alpha = beta + 5
X = Pareto('x', xm, alpha)
dens = density(X)
x = Symbol('x')
assert dens(x) == x**(-(alpha + 1))*alpha*xm**alpha
def test_uniformsum_d():
n = Symbol("n", integer=True)
k = Symbol("k")
X = UniformSum('x', n)
d = density(X)(x)
assert d == 1/factorial(n - 1)*Sum((-1)**k*(x - k)**(n - 1) *
binomial(n, k), (k, 0, floor(x)))
def test_raised_cosine():
mu = Symbol("mu", extended_real=True)
s = Symbol("s", positive=True)
X = RaisedCosine("x", mu, s)
assert density(X)(x) == (Piecewise(((cos(pi*(x - mu)/s) + 1)/(2*s),
And(x <= mu + s, mu - s <= x)), (0, True)))
assert X.pspace.domain.set == Interval(mu - s, mu + s)
def test_die_args():
pytest.raises(ValueError, lambda: Die('X', -1)) # issue sympy/sympy#8105: negative sides.
pytest.raises(ValueError, lambda: Die('X', 0))
pytest.raises(ValueError, lambda: Die('X', 1.5)) # issue sympy/sympy#8103: non integer sides.
k = Symbol('k')
sym_die = Die('X', k)
pytest.raises(ValueError, lambda: density(sym_die).dict)
def test_FiniteRV():
F = FiniteRV('F', {1: Rational(1, 2), 2: Rational(1, 4), 3: Rational(1, 4)})
assert dict(density(F).items()) == {1: Rational(1, 2), 2: Rational(1, 4), 3: Rational(1, 4)}
assert P(F >= 2) == Rational(1, 2)
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
def test_density_call():
x = Bernoulli('x', p)
d = density(x)
assert d(0) == 1 - p
assert d(5) == 0
assert 0 in d
assert 5 not in d
assert d(0) == d[0]
def test_GeometricDistribution():
p = Rational(1, 5)
d = GeometricDistribution(p)
assert d.expectation(x, x) == 1/p
assert d.expectation(x**2, x) - d.expectation(x, x)**2 == (1-p)/p**2
assert abs(d.cdf(20000).evalf() - 1) < .001
X = Geometric("x", p)
assert density(X)(z) == Rational(4, 5)**(z - 1)/5
def test_benini():
alpha = Symbol("alpha", positive=True)
b = Symbol("beta", positive=True)
sigma = Symbol("sigma", positive=True)
X = Benini('x', alpha, b, sigma)
assert density(X)(x) == ((alpha/x + 2*b*log(x/sigma)/x)
* exp(-alpha*log(x/sigma) - b*log(x/sigma)**2))
assert X.pspace.domain.set == Interval(sigma, oo, False, True)
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_uniformsum():
n = Symbol("n", integer=True)
X = UniformSum('x', n)
assert X.pspace.domain.set == Interval(0, n)
# see test_uniformsum_d()
d = density(X)(z)
assert str(d) == ("Sum((-1)**_k*(z - _k)**(n - 1)*binomial(n, _k), "
"(_k, 0, floor(z)))/factorial(n - 1)")
def test_triangular():
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
X = Triangular('x', a, b, c)
assert density(X)(x) == Piecewise(
((2 * x - 2 * a) / ((-a + b) * (-a + c)), And(a <= x, x < c)),
(2 / (-a + b), x == c),
((-2 * x + 2 * b) / ((-a + b) * (b - c)), And(x <= b, c < x)),
(0, True))
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_triangular():
a = Symbol("a")
b = Symbol("b")
c = Symbol("c")
X = Triangular('x', a, b, c)
assert density(X)(x) == Piecewise(
((2*x - 2*a)/((-a + b)*(-a + c)), And(a <= x, x < c)),
(2/(-a + b), Eq(x, c)),
((-2*x + 2*b)/((-a + b)*(b - c)), And(x <= b, c < x)),
(0, True))
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 + S.Half) / gamma(mu + 1))
assert simplify(variance(
X, meijerg=True)) == (omega - omega * gamma(mu + Rational(1, 2))**2 /
(gamma(mu) * gamma(mu + 1)))
def test_die_args():
pytest.raises(
ValueError,
lambda: Die('X', -1)) # issue sympy/sympy#8105: negative sides.
pytest.raises(ValueError, lambda: Die('X', 0))
pytest.raises(
ValueError,
lambda: Die('X', 1.5)) # issue sympy/sympy#8103: non integer sides.
k = Symbol('k')
sym_die = Die('X', k)
pytest.raises(ValueError, lambda: density(sym_die).dict)
def test_FiniteRV():
F = FiniteRV('F', {1: S.Half, 2: S.One / 4, 3: S.One / 4})
assert dict(density(F).items()) == {
Integer(1): S.Half,
Integer(2): S.One / 4,
Integer(3): S.One / 4
}
assert P(F >= 2) == S.Half
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
def test_coins():
C, D = Coin('C'), Coin('D')
H, T = symbols('H, T')
assert P(Eq(C, D)) == Rational(1, 2)
assert density(Tuple(C, D)) == {
(H, H): Rational(1, 4),
(H, T): Rational(1, 4),
(T, H): Rational(1, 4),
(T, T): Rational(1, 4)
}
assert dict(density(C).items()) == {H: Rational(1, 2), T: Rational(1, 2)}
F = Coin('F', Rational(1, 10))
assert P(Eq(F, H)) == Rational(1, 10)
d = pspace(C).domain
assert d.as_boolean() == Or(Eq(C.symbol, H), Eq(C.symbol, T))
pytest.raises(ValueError,
lambda: P(C > D)) # Can't intelligently compare H to T
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_discreteuniform():
# Symbolic
a, b, c = symbols('a b c')
X = DiscreteUniform('X', [a, b, c])
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_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_FiniteRV():
F = FiniteRV('F', {
1: Rational(1, 2),
2: Rational(1, 4),
3: Rational(1, 4)
})
assert dict(density(F).items()) == {
1: Rational(1, 2),
2: Rational(1, 4),
3: Rational(1, 4)
}
assert P(F >= 2) == Rational(1, 2)
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
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_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_Density():
X = Die('X', 6)
d = Density(X)
assert d.doit() == density(X)
def test_cauchy():
x0 = Symbol('x0')
gamma = Symbol('gamma', positive=True)
X = Cauchy('x', x0, gamma)
assert density(X)(x) == 1/(pi*gamma*(1 + (x - x0)**2/gamma**2))
def test_normality():
X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
x, z = symbols('x, z', real=True)
dens = density(X - Y, Eq(X + Y, z))
assert integrate(dens(x), (x, -oo, oo)) == 1
def test_chi_squared():
k = Symbol('k', integer=True)
X = ChiSquared('x', k)
assert density(X)(x) == 2**(-k/2)*x**(k/2 - 1)*exp(-x/2)/gamma(k/2)
def test_chi():
k = Symbol('k', integer=True)
X = Chi('x', k)
assert density(X)(x) == 2**(-k/2 + 1)*x**(k - 1)*exp(-x**2/2)/gamma(k/2)
def test_gamma_inverse():
a = Symbol('a', positive=True)
b = Symbol('b', positive=True)
X = GammaInverse('x', a, b)
assert density(X)(x) == x**(-a - 1)*b**a*exp(-b/x)/gamma(a)
def test_erlang():
k = Symbol('k', integer=True, positive=True)
l = Symbol('l', positive=True)
X = Erlang('x', k, l)
assert density(X)(x) == x**(k - 1)*l**k*exp(-x*l)/gamma(k)
def test_laplace():
mu = Symbol('mu')
b = Symbol('b', positive=True)
X = Laplace('x', mu, b)
assert density(X)(x) == exp(-abs(x - mu)/b)/(2*b)
def test_kumaraswamy():
a = Symbol('a', positive=True)
b = Symbol('b', positive=True)
X = Kumaraswamy('x', a, b)
assert density(X)(x) == x**(a - 1)*a*b*(-x**a + 1)**(b - 1)
def test_studentt():
nu = Symbol('nu', positive=True)
X = StudentT('x', nu)
assert density(X)(x) == (1 + x**2/nu)**(-nu/2 - 1/2)/(sqrt(nu)*beta(1/2, nu/2))
def test_logistic():
mu = Symbol('mu', extended_real=True)
s = Symbol('s', positive=True)
X = Logistic('x', mu, s)
assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
def test_conjugate_priors():
mu = Normal('mu', 2, 3)
x = Normal('x', mu, 1)
assert isinstance(simplify(density(mu, Eq(x, y), evaluate=False)(z)),
Integral)
def test_random_parameters():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert density(meas, evaluate=False)(z)
assert isinstance(pspace(meas), ProductPSpace)
def test_density_unevaluated():
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 2)
assert isinstance(density(X+Y, evaluate=False)(z), Integral)
def test_wignersemicircle():
R = Symbol('R', positive=True)
X = WignerSemicircle('x', R)
assert density(X)(x) == 2*sqrt(-x**2 + R**2)/(pi*R**2)
assert E(X) == 0
def test_von_mises():
mu = Symbol('mu')
k = Symbol('k', positive=True)
X = VonMises('x', mu, k)
assert density(X)(x) == exp(k*cos(x - mu))/(2*pi*besseli(0, k))
def test_density_constant():
assert density(3)(2) == 0
assert density(3)(3) == DiracDelta(0)
def test_betaprime():
alpha = Symbol('alpha', positive=True)
betap = Symbol('beta', positive=True)
X = BetaPrime('x', alpha, betap)
assert density(X)(x) == x**(alpha - 1)*(x + 1)**(-alpha - betap)/beta(alpha, betap)