phivenv/Lib/site-packages/sympy/stats/tests/test_compound_rv.py

from sympy.concrete.summations import Sum
from sympy.core.numbers import (oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.error_functions import erf
from sympy.functions.special.gamma_functions import gamma
from sympy.integrals.integrals import Integral
from sympy.sets.sets import Interval
from sympy.stats import (Normal, P, E, density, Gamma, Poisson, Rayleigh,
                        variance, Bernoulli, Beta, Uniform, cdf)
from sympy.stats.compound_rv import CompoundDistribution, CompoundPSpace
from sympy.stats.crv_types import NormalDistribution
from sympy.stats.drv_types import PoissonDistribution
from sympy.stats.frv_types import BernoulliDistribution
from sympy.testing.pytest import raises, ignore_warnings
from sympy.stats.joint_rv_types import MultivariateNormalDistribution

from sympy.abc import x


# helpers for testing troublesome unevaluated expressions
flat = lambda s: ''.join(str(s).split())
streq = lambda *a: len(set(map(flat, a))) == 1
assert streq(x, x)
assert streq(x, 'x')
assert not streq(x, x + 1)


def test_normal_CompoundDist():
    X = Normal('X', 1, 2)
    Y = Normal('X', X, 4)
    assert density(Y)(x).simplify() == sqrt(10)*exp(-x**2/40 + x/20 - S(1)/40)/(20*sqrt(pi))
    assert E(Y) == 1 # it is always equal to mean of X
    assert P(Y > 1) == S(1)/2 # as 1 is the mean
    assert P(Y > 5).simplify() ==  S(1)/2 - erf(sqrt(10)/5)/2
    assert variance(Y) == variance(X) + 4**2 # 2**2 + 4**2
    # https://math.stackexchange.com/questions/1484451/
    # (Contains proof of E and variance computation)


def test_poisson_CompoundDist():
    k, t, y = symbols('k t y', positive=True, real=True)
    G = Gamma('G', k, t)
    D = Poisson('P', G)
    assert density(D)(y).simplify() == t**y*(t + 1)**(-k - y)*gamma(k + y)/(gamma(k)*gamma(y + 1))
    # https://en.wikipedia.org/wiki/Negative_binomial_distribution#Gamma%E2%80%93Poisson_mixture
    assert E(D).simplify() == k*t # mean of NegativeBinomialDistribution


def test_bernoulli_CompoundDist():
    X = Beta('X', 1, 2)
    Y = Bernoulli('Y', X)
    assert density(Y).dict == {0: S(2)/3, 1: S(1)/3}
    assert E(Y) == P(Eq(Y, 1)) == S(1)/3
    assert variance(Y) == S(2)/9
    assert cdf(Y) == {0: S(2)/3, 1: 1}

    # test issue 8128
    a = Bernoulli('a', S(1)/2)
    b = Bernoulli('b', a)
    assert density(b).dict == {0: S(1)/2, 1: S(1)/2}
    assert P(b > 0.5) == S(1)/2

    X = Uniform('X', 0, 1)
    Y = Bernoulli('Y', X)
    assert E(Y) == S(1)/2
    assert P(Eq(Y, 1)) == E(Y)


def test_unevaluated_CompoundDist():
    # these tests need to be removed once they work with evaluation as they are currently not
    # evaluated completely in sympy.
    R = Rayleigh('R', 4)
    X = Normal('X', 3, R)
    ans = '''
        Piecewise(((-sqrt(pi)*sinh(x/4 - 3/4) + sqrt(pi)*cosh(x/4 - 3/4))/(
        8*sqrt(pi)), Abs(arg(x - 3)) <= pi/4), (Integral(sqrt(2)*exp(-(x - 3)
        **2/(2*R**2))*exp(-R**2/32)/(32*sqrt(pi)), (R, 0, oo)), True))'''
    assert streq(density(X)(x), ans)

    expre = '''
        Integral(X*Integral(sqrt(2)*exp(-(X-3)**2/(2*R**2))*exp(-R**2/32)/(32*
        sqrt(pi)),(R,0,oo)),(X,-oo,oo))'''
    with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
        assert streq(E(X, evaluate=False).rewrite(Integral), expre)

    X = Poisson('X', 1)
    Y = Poisson('Y', X)
    Z = Poisson('Z', Y)
    exprd = Sum(exp(-Y)*Y**x*Sum(exp(-1)*exp(-X)*X**Y/(factorial(X)*factorial(Y)
                ), (X, 0, oo))/factorial(x), (Y, 0, oo))
    assert density(Z)(x) == exprd

    N = Normal('N', 1, 2)
    M = Normal('M', 3, 4)
    D = Normal('D', M, N)
    exprd = '''
        Integral(sqrt(2)*exp(-(N-1)**2/8)*Integral(exp(-(x-M)**2/(2*N**2))*exp
        (-(M-3)**2/32)/(8*pi*N),(M,-oo,oo))/(4*sqrt(pi)),(N,-oo,oo))'''
    assert streq(density(D, evaluate=False)(x), exprd)


def test_Compound_Distribution():
    X = Normal('X', 2, 4)
    N = NormalDistribution(X, 4)
    C = CompoundDistribution(N)
    assert C.is_Continuous
    assert C.set == Interval(-oo, oo)
    assert C.pdf(x, evaluate=True).simplify() == exp(-x**2/64 + x/16 - S(1)/16)/(8*sqrt(pi))

    assert not isinstance(CompoundDistribution(NormalDistribution(2, 3)),
                            CompoundDistribution)
    M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]])
    raises(NotImplementedError, lambda: CompoundDistribution(M))

    X = Beta('X', 2, 4)
    B = BernoulliDistribution(X, 1, 0)
    C = CompoundDistribution(B)
    assert C.is_Finite
    assert C.set == {0, 1}
    y = symbols('y', negative=False, integer=True)
    assert C.pdf(y, evaluate=True) == Piecewise((S(1)/(30*beta(2, 4)), Eq(y, 0)),
                (S(1)/(60*beta(2, 4)), Eq(y, 1)), (0, True))

    k, t, z = symbols('k t z', positive=True, real=True)
    G = Gamma('G', k, t)
    X = PoissonDistribution(G)
    C = CompoundDistribution(X)
    assert C.is_Discrete
    assert C.set == S.Naturals0
    assert C.pdf(z, evaluate=True).simplify() == t**z*(t + 1)**(-k - z)*gamma(k \
                    + z)/(gamma(k)*gamma(z + 1))


def test_compound_pspace():
    X = Normal('X', 2, 4)
    Y = Normal('Y', 3, 6)
    assert not isinstance(Y.pspace, CompoundPSpace)
    N = NormalDistribution(1, 2)
    D = PoissonDistribution(3)
    B = BernoulliDistribution(0.2, 1, 0)
    pspace1 = CompoundPSpace('N', N)
    pspace2 = CompoundPSpace('D', D)
    pspace3 = CompoundPSpace('B', B)
    assert not isinstance(pspace1, CompoundPSpace)
    assert not isinstance(pspace2, CompoundPSpace)
    assert not isinstance(pspace3, CompoundPSpace)
    M = MultivariateNormalDistribution([1, 2], [[2, 1], [1, 2]])
    raises(ValueError, lambda: CompoundPSpace('M', M))
    Y = Normal('Y', X, 6)
    assert isinstance(Y.pspace, CompoundPSpace)
    assert Y.pspace.distribution == CompoundDistribution(NormalDistribution(X, 6))
    assert Y.pspace.domain.set == Interval(-oo, oo)