def test_characteristic_function(): X = Uniform('x', 0, 1) cf = characteristic_function(X) assert cf(1) == -I * (-1 + exp(I)) Y = Normal('y', 1, 1) cf = characteristic_function(Y) assert cf(0) == 1 assert cf(1) == exp(I - S.Half) Z = Exponential('z', 5) cf = characteristic_function(Z) assert cf(0) == 1 assert cf(1).expand() == Rational(25, 26) + I * Rational(5, 26) X = GaussianInverse('x', 1, 1) cf = characteristic_function(X) assert cf(0) == 1 assert cf(1) == exp(1 - sqrt(1 - 2 * I)) X = ExGaussian('x', 0, 1, 1) cf = characteristic_function(X) assert cf(0) == 1 assert cf(1) == (1 + I) * exp(Rational(-1, 2)) / 2 L = Levy('x', 0, 1) cf = characteristic_function(L) assert cf(0) == 1 assert cf(1) == exp(-sqrt(2) * sqrt(-I))
def test_characteristic_function(): X = Uniform('x', 0, 1) cf = characteristic_function(X) assert cf(1) == -I * (-1 + exp(I)) Y = Normal('y', 1, 1) cf = characteristic_function(Y) assert cf(0) == 1 assert cf(1) == exp(I - S(1) / 2) Z = Exponential('z', 5) cf = characteristic_function(Z) assert cf(0) == 1 assert cf(1).expand() == S(25) / 26 + 5 * I / 26 X = GaussianInverse('x', 1, 1) cf = characteristic_function(X) assert cf(0) == 1 assert cf(1) == exp(1 - sqrt(1 - 2 * I)) X = ExGaussian('x', 0, 1, 1) cf = characteristic_function(X) assert cf(0) == 1 assert cf(1) == (1 + I) * exp(-S(1) / 2) / 2
def test_exgaussian(): m, z = symbols("m, z") s, l = symbols("s, l", positive=True) X = ExGaussian("x", m, s, l) assert density(X)(z) == l*exp(l*(l*s**2 + 2*m - 2*z)/2) *\ erfc(sqrt(2)*(l*s**2 + m - z)/(2*s))/2 # Note: actual_output simplifies to expected_output. # Ideally cdf(X)(z) would return expected_output # expected_output = (erf(sqrt(2)*(l*s**2 + m - z)/(2*s)) - 1)*exp(l*(l*s**2 + 2*m - 2*z)/2)/2 - erf(sqrt(2)*(m - z)/(2*s))/2 + S.Half u = l * (z - m) v = l * s GaussianCDF1 = cdf(Normal('x', 0, v))(u) GaussianCDF2 = cdf(Normal('x', v**2, v))(u) actual_output = GaussianCDF1 - exp(-u + (v**2 / 2) + log(GaussianCDF2)) assert cdf(X)(z) == actual_output # assert simplify(actual_output) == expected_output assert variance(X).expand() == s**2 + l**(-2) assert skewness(X).expand() == 2 / (l**3 * s**2 * sqrt(s**2 + l**(-2)) + l * sqrt(s**2 + l**(-2)))
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)