Exemple #1
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def test_logistic():
    mu = Symbol("mu", real=True)
    s = Symbol("s", positive=True)
    p = Symbol("p", positive=True)

    X = Logistic('x', mu, s)
    assert density(X)(x) == exp((-x + mu) / s) / (s * (exp(
        (-x + mu) / s) + 1)**2)
    assert cdf(X)(x) == 1 / (exp((mu - x) / s) + 1)
    assert quantile(X)(p) == mu - s * log(-S.One + 1 / p)
Exemple #2
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def test_logistic():
    mu = Symbol("mu", real=True)
    s = Symbol("s", positive=True)
    p = Symbol("p", positive=True)

    X = Logistic('x', mu, s)

    #Tests characteristics_function
    assert characteristic_function(X)(x) == \
           (Piecewise((pi*s*x*exp(I*mu*x)/sinh(pi*s*x), Ne(x, 0)), (1, True)))

    assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
    assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1)
    assert quantile(X)(p) == mu - s*log(-S.One + 1/p)
Exemple #3
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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)
Exemple #4
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def test_logistic():
    mu = Symbol("mu", 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)
Exemple #5
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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)