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)
assert cdf(X)(x) == Piecewise((lowergamma(k, l * x) / gamma(k), x > 0),
(0, True))
def test_long_precomputed_cdf():
x = symbols("x", real=True, finite=True)
distribs = [
Arcsin("A", -5, 9),
Dagum("D", 4, 10, 3),
Erlang("E", 14, 5),
Frechet("F", 2, 6, -3),
Gamma("G", 2, 7),
GammaInverse("GI", 3, 5),
Kumaraswamy("K", 6, 8),
Laplace("LA", -5, 4),
Logistic("L", -6, 7),
Nakagami("N", 2, 7),
StudentT("S", 4)
]
for distr in distribs:
for _ in range(5):
assert tn(diff(cdf(distr)(x), x), density(distr)(x), x, a=0, b=0, c=1, d=0)
US = UniformSum("US", 5)
pdf01 = density(US)(x).subs(floor(x), 0).doit() # pdf on (0, 1)
cdf01 = cdf(US, evaluate=False)(x).subs(floor(x), 0).doit() # cdf on (0, 1)
assert tn(diff(cdf01, x), pdf01, x, a=0, b=0, c=1, d=0)
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_moment_generating_function():
t = symbols('t', positive=True)
# Symbolic tests
a, b, c = symbols('a b c')
mgf = moment_generating_function(Beta('x', a, b))(t)
assert mgf == hyper((a, ), (a + b, ), t)
mgf = moment_generating_function(Chi('x', a))(t)
assert mgf == sqrt(2)*t*gamma(a/2 + S.Half)*\
hyper((a/2 + S.Half,), (Rational(3, 2),), t**2/2)/gamma(a/2) +\
hyper((a/2,), (S.Half,), t**2/2)
mgf = moment_generating_function(ChiSquared('x', a))(t)
assert mgf == (1 - 2 * t)**(-a / 2)
mgf = moment_generating_function(Erlang('x', a, b))(t)
assert mgf == (1 - t / b)**(-a)
mgf = moment_generating_function(ExGaussian("x", a, b, c))(t)
assert mgf == exp(a * t + b**2 * t**2 / 2) / (1 - t / c)
mgf = moment_generating_function(Exponential('x', a))(t)
assert mgf == a / (a - t)
mgf = moment_generating_function(Gamma('x', a, b))(t)
assert mgf == (-b * t + 1)**(-a)
mgf = moment_generating_function(Gumbel('x', a, b))(t)
assert mgf == exp(b * t) * gamma(-a * t + 1)
mgf = moment_generating_function(Gompertz('x', a, b))(t)
assert mgf == b * exp(b) * expint(t / a, b)
mgf = moment_generating_function(Laplace('x', a, b))(t)
assert mgf == exp(a * t) / (-b**2 * t**2 + 1)
mgf = moment_generating_function(Logistic('x', a, b))(t)
assert mgf == exp(a * t) * beta(-b * t + 1, b * t + 1)
mgf = moment_generating_function(Normal('x', a, b))(t)
assert mgf == exp(a * t + b**2 * t**2 / 2)
mgf = moment_generating_function(Pareto('x', a, b))(t)
assert mgf == b * (-a * t)**b * uppergamma(-b, -a * t)
mgf = moment_generating_function(QuadraticU('x', a, b))(t)
assert str(mgf) == (
"(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - "
"3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)")
mgf = moment_generating_function(RaisedCosine('x', a, b))(t)
assert mgf == pi**2 * exp(a * t) * sinh(b * t) / (b * t *
(b**2 * t**2 + pi**2))
mgf = moment_generating_function(Rayleigh('x', a))(t)
assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\
*exp(a**2*t**2/2)/2 + 1
mgf = moment_generating_function(Triangular('x', a, b, c))(t)
assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + "
"2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))")
mgf = moment_generating_function(Uniform('x', a, b))(t)
assert mgf == (-exp(a * t) + exp(b * t)) / (t * (-a + b))
mgf = moment_generating_function(UniformSum('x', a))(t)
assert mgf == ((exp(t) - 1) / t)**a
mgf = moment_generating_function(WignerSemicircle('x', a))(t)
assert mgf == 2 * besseli(1, a * t) / (a * t)
# Numeric tests
mgf = moment_generating_function(Beta('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == hyper((2, ), (3, ), 1) / 2
mgf = moment_generating_function(Chi('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == sqrt(2) * hyper(
(1, ), (Rational(3, 2), ), S.Half) / sqrt(pi) + hyper(
(Rational(3, 2), ),
(Rational(3, 2), ), S.Half) + 2 * sqrt(2) * hyper(
(2, ), (Rational(5, 2), ), S.Half) / (3 * sqrt(pi))
mgf = moment_generating_function(ChiSquared('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == I
mgf = moment_generating_function(Erlang('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(ExGaussian("x", 0, 1, 1))(t)
assert mgf.diff(t).subs(t, 2) == -exp(2)
mgf = moment_generating_function(Exponential('x', 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Gamma('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Gumbel('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == EulerGamma + 1
mgf = moment_generating_function(Gompertz('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == -e * meijerg(((), (1, 1)),
((0, 0, 0), ()), 1)
mgf = moment_generating_function(Laplace('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == 1
mgf = moment_generating_function(Logistic('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == beta(1, 1)
mgf = moment_generating_function(Normal('x', 0, 1))(t)
assert mgf.diff(t).subs(t, 1) == exp(S.Half)
mgf = moment_generating_function(Pareto('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 0) == expint(1, 0)
mgf = moment_generating_function(QuadraticU('x', 1, 2))(t)
assert mgf.diff(t).subs(t, 1) == -12 * e - 3 * exp(2)
mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t)
assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\
(1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2)
mgf = moment_generating_function(Rayleigh('x', 1))(t)
assert mgf.diff(t).subs(t, 0) == sqrt(2) * sqrt(pi) / 2
mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t)
assert mgf.diff(t).subs(t, 1) == -e + exp(3)
mgf = moment_generating_function(Uniform('x', 0, 1))(t)
assert mgf.diff(t).subs(t, 1) == 1
mgf = moment_generating_function(UniformSum('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == 1
mgf = moment_generating_function(WignerSemicircle('x', 1))(t)
assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\
besseli(0, 1)
