I am done

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

@@ -0,0 +1,159 @@
+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)

rl/Lib/site-packages/sympy/stats/tests/test_discrete_rv.py

@@ -0,0 +1,297 @@
+from sympy.concrete.summations import Sum
+from sympy.core.numbers import (I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.complexes import (im, re)
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.integers import floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.bessel import besseli
+from sympy.functions.special.beta_functions import beta
+from sympy.functions.special.zeta_functions import zeta
+from sympy.sets.sets import FiniteSet
+from sympy.simplify.simplify import simplify
+from sympy.utilities.lambdify import lambdify
+from sympy.core.relational import Eq, Ne
+from sympy.functions.elementary.exponential import exp
+from sympy.logic.boolalg import Or
+from sympy.sets.fancysets import Range
+from sympy.stats import (P, E, variance, density, characteristic_function,
+                         where, moment_generating_function, skewness, cdf,
+                         kurtosis, coskewness)
+from sympy.stats.drv_types import (PoissonDistribution, GeometricDistribution,
+                                   FlorySchulz, Poisson, Geometric, Hermite, Logarithmic,
+                                    NegativeBinomial, Skellam, YuleSimon, Zeta,
+                                    DiscreteRV)
+from sympy.testing.pytest import slow, nocache_fail, raises
+from sympy.stats.symbolic_probability import Expectation
+
+x = Symbol('x')
+
+
+def test_PoissonDistribution():
+    l = 3
+    p = PoissonDistribution(l)
+    assert abs(p.cdf(10).evalf() - 1) < .001
+    assert abs(p.cdf(10.4).evalf() - 1) < .001
+    assert p.expectation(x, x) == l
+    assert p.expectation(x**2, x) - p.expectation(x, x)**2 == l
+
+
+def test_Poisson():
+    l = 3
+    x = Poisson('x', l)
+    assert E(x) == l
+    assert E(2*x) == 2*l
+    assert variance(x) == l
+    assert density(x) == PoissonDistribution(l)
+    assert isinstance(E(x, evaluate=False), Expectation)
+    assert isinstance(E(2*x, evaluate=False), Expectation)
+    # issue 8248
+    assert x.pspace.compute_expectation(1) == 1
+
+
+def test_FlorySchulz():
+    a = Symbol("a")
+    z = Symbol("z")
+    x = FlorySchulz('x', a)
+    assert E(x) == (2 - a)/a
+    assert (variance(x) - 2*(1 - a)/a**2).simplify() == S(0)
+    assert density(x)(z) == a**2*z*(1 - a)**(z - 1)
+
+
+@slow
+def test_GeometricDistribution():
+    p = S.One / 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
+    assert abs(d.cdf(20000.8).evalf() - 1) < .001
+    G = Geometric('G', p=S(1)/4)
+    assert cdf(G)(S(7)/2) == P(G <= S(7)/2)
+
+    X = Geometric('X', Rational(1, 5))
+    Y = Geometric('Y', Rational(3, 10))
+    assert coskewness(X, X + Y, X + 2*Y).simplify() == sqrt(230)*Rational(81, 1150)
+
+
+def test_Hermite():
+    a1 = Symbol("a1", positive=True)
+    a2 = Symbol("a2", negative=True)
+    raises(ValueError, lambda: Hermite("H", a1, a2))
+
+    a1 = Symbol("a1", negative=True)
+    a2 = Symbol("a2", positive=True)
+    raises(ValueError, lambda: Hermite("H", a1, a2))
+
+    a1 = Symbol("a1", positive=True)
+    x = Symbol("x")
+    H = Hermite("H", a1, a2)
+    assert moment_generating_function(H)(x) == exp(a1*(exp(x) - 1)
+                                            + a2*(exp(2*x) - 1))
+    assert characteristic_function(H)(x) == exp(a1*(exp(I*x) - 1)
+                                            + a2*(exp(2*I*x) - 1))
+    assert E(H) == a1 + 2*a2
+
+    H = Hermite("H", a1=5, a2=4)
+    assert density(H)(2) == 33*exp(-9)/2
+    assert E(H) == 13
+    assert variance(H) == 21
+    assert kurtosis(H) == Rational(464,147)
+    assert skewness(H) == 37*sqrt(21)/441
+
+def test_Logarithmic():
+    p = S.Half
+    x = Logarithmic('x', p)
+    assert E(x) == -p / ((1 - p) * log(1 - p))
+    assert variance(x) == -1/log(2)**2 + 2/log(2)
+    assert E(2*x**2 + 3*x + 4) == 4 + 7 / log(2)
+    assert isinstance(E(x, evaluate=False), Expectation)
+
+
+@nocache_fail
+def test_negative_binomial():
+    r = 5
+    p = S.One / 3
+    x = NegativeBinomial('x', r, p)
+    assert E(x) == p*r / (1-p)
+    # This hangs when run with the cache disabled:
+    assert variance(x) == p*r / (1-p)**2
+    assert E(x**5 + 2*x + 3) == Rational(9207, 4)
+    assert isinstance(E(x, evaluate=False), Expectation)
+
+
+def test_skellam():
+    mu1 = Symbol('mu1')
+    mu2 = Symbol('mu2')
+    z = Symbol('z')
+    X = Skellam('x', mu1, mu2)
+
+    assert density(X)(z) == (mu1/mu2)**(z/2) * \
+        exp(-mu1 - mu2)*besseli(z, 2*sqrt(mu1*mu2))
+    assert skewness(X).expand() == mu1/(mu1*sqrt(mu1 + mu2) + mu2 *
+                sqrt(mu1 + mu2)) - mu2/(mu1*sqrt(mu1 + mu2) + mu2*sqrt(mu1 + mu2))
+    assert variance(X).expand() == mu1 + mu2
+    assert E(X) == mu1 - mu2
+    assert characteristic_function(X)(z) == exp(
+        mu1*exp(I*z) - mu1 - mu2 + mu2*exp(-I*z))
+    assert moment_generating_function(X)(z) == exp(
+        mu1*exp(z) - mu1 - mu2 + mu2*exp(-z))
+
+
+def test_yule_simon():
+    from sympy.core.singleton import S
+    rho = S(3)
+    x = YuleSimon('x', rho)
+    assert simplify(E(x)) == rho / (rho - 1)
+    assert simplify(variance(x)) == rho**2 / ((rho - 1)**2 * (rho - 2))
+    assert isinstance(E(x, evaluate=False), Expectation)
+    # To test the cdf function
+    assert cdf(x)(x) == Piecewise((-beta(floor(x), 4)*floor(x) + 1, x >= 1), (0, True))
+
+
+def test_zeta():
+    s = S(5)
+    x = Zeta('x', s)
+    assert E(x) == zeta(s-1) / zeta(s)
+    assert simplify(variance(x)) == (
+        zeta(s) * zeta(s-2) - zeta(s-1)**2) / zeta(s)**2
+
+
+def test_discrete_probability():
+    X = Geometric('X', Rational(1, 5))
+    Y = Poisson('Y', 4)
+    G = Geometric('e', x)
+    assert P(Eq(X, 3)) == Rational(16, 125)
+    assert P(X < 3) == Rational(9, 25)
+    assert P(X > 3) == Rational(64, 125)
+    assert P(X >= 3) == Rational(16, 25)
+    assert P(X <= 3) == Rational(61, 125)
+    assert P(Ne(X, 3)) == Rational(109, 125)
+    assert P(Eq(Y, 3)) == 32*exp(-4)/3
+    assert P(Y < 3) == 13*exp(-4)
+    assert P(Y > 3).equals(32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3)
+    assert P(Y >= 3).equals(32*(Rational(-39, 32) + 3*exp(4)/32)*exp(-4)/3)
+    assert P(Y <= 3) == 71*exp(-4)/3
+    assert P(Ne(Y, 3)).equals(
+        13*exp(-4) + 32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3)
+    assert P(X < S.Infinity) is S.One
+    assert P(X > S.Infinity) is S.Zero
+    assert P(G < 3) == x*(2-x)
+    assert P(Eq(G, 3)) == x*(-x + 1)**2
+
+
+def test_DiscreteRV():
+    p = S(1)/2
+    x = Symbol('x', integer=True, positive=True)
+    pdf = p*(1 - p)**(x - 1) # pdf of Geometric Distribution
+    D = DiscreteRV(x, pdf, set=S.Naturals, check=True)
+    assert E(D) == E(Geometric('G', S(1)/2)) == 2
+    assert P(D > 3) == S(1)/8
+    assert D.pspace.domain.set == S.Naturals
+    raises(ValueError, lambda: DiscreteRV(x, x, FiniteSet(*range(4)), check=True))
+
+    # purposeful invalid pmf but it should not raise since check=False
+    # see test_drv_types.test_ContinuousRV for explanation
+    X = DiscreteRV(x, 1/x, S.Naturals)
+    assert P(X < 2) == 1
+    assert E(X) == oo
+
+def test_precomputed_characteristic_functions():
+    import mpmath
+
+    def test_cf(dist, support_lower_limit, support_upper_limit):
+        pdf = density(dist)
+        t = S('t')
+        x = S('x')
+
+        # first function is the hardcoded CF of the distribution
+        cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath')
+
+        # second function is the Fourier transform of the density function
+        f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath')
+        cf2 = lambda t: mpmath.nsum(lambda x: f(x, t), [
+            support_lower_limit, support_upper_limit], maxdegree=10)
+
+        # compare the two functions at various points
+        for test_point in [2, 5, 8, 11]:
+            n1 = cf1(test_point)
+            n2 = cf2(test_point)
+
+            assert abs(re(n1) - re(n2)) < 1e-12
+            assert abs(im(n1) - im(n2)) < 1e-12
+
+    test_cf(Geometric('g', Rational(1, 3)), 1, mpmath.inf)
+    test_cf(Logarithmic('l', Rational(1, 5)), 1, mpmath.inf)
+    test_cf(NegativeBinomial('n', 5, Rational(1, 7)), 0, mpmath.inf)
+    test_cf(Poisson('p', 5), 0, mpmath.inf)
+    test_cf(YuleSimon('y', 5), 1, mpmath.inf)
+    test_cf(Zeta('z', 5), 1, mpmath.inf)
+
+
+def test_moment_generating_functions():
+    t = S('t')
+
+    geometric_mgf = moment_generating_function(Geometric('g', S.Half))(t)
+    assert geometric_mgf.diff(t).subs(t, 0) == 2
+
+    logarithmic_mgf = moment_generating_function(Logarithmic('l', S.Half))(t)
+    assert logarithmic_mgf.diff(t).subs(t, 0) == 1/log(2)
+
+    negative_binomial_mgf = moment_generating_function(
+        NegativeBinomial('n', 5, Rational(1, 3)))(t)
+    assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2)
+
+    poisson_mgf = moment_generating_function(Poisson('p', 5))(t)
+    assert poisson_mgf.diff(t).subs(t, 0) == 5
+
+    skellam_mgf = moment_generating_function(Skellam('s', 1, 1))(t)
+    assert skellam_mgf.diff(t).subs(
+        t, 2) == (-exp(-2) + exp(2))*exp(-2 + exp(-2) + exp(2))
+
+    yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t)
+    assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2)
+
+    zeta_mgf = moment_generating_function(Zeta('z', 5))(t)
+    assert zeta_mgf.diff(t).subs(t, 0) == pi**4/(90*zeta(5))
+
+
+def test_Or():
+    X = Geometric('X', S.Half)
+    assert P(Or(X < 3, X > 4)) == Rational(13, 16)
+    assert P(Or(X > 2, X > 1)) == P(X > 1)
+    assert P(Or(X >= 3, X < 3)) == 1
+
+
+def test_where():
+    X = Geometric('X', Rational(1, 5))
+    Y = Poisson('Y', 4)
+    assert where(X**2 > 4).set == Range(3, S.Infinity, 1)
+    assert where(X**2 >= 4).set == Range(2, S.Infinity, 1)
+    assert where(Y**2 < 9).set == Range(0, 3, 1)
+    assert where(Y**2 <= 9).set == Range(0, 4, 1)
+
+
+def test_conditional():
+    X = Geometric('X', Rational(2, 3))
+    Y = Poisson('Y', 3)
+    assert P(X > 2, X > 3) == 1
+    assert P(X > 3, X > 2) == Rational(1, 3)
+    assert P(Y > 2, Y < 2) == 0
+    assert P(Eq(Y, 3), Y >= 0) == 9*exp(-3)/2
+    assert P(Eq(Y, 3), Eq(Y, 2)) == 0
+    assert P(X < 2, Eq(X, 2)) == 0
+    assert P(X > 2, Eq(X, 3)) == 1
+
+
+def test_product_spaces():
+    X1 = Geometric('X1', S.Half)
+    X2 = Geometric('X2', Rational(1, 3))
+    assert str(P(X1 + X2 < 3).rewrite(Sum)) == (
+        "Sum(Piecewise((1/(4*2**n), n >= -1), (0, True)), (n, -oo, -1))/3")
+    assert str(P(X1 + X2 > 3).rewrite(Sum)) == (
+        'Sum(Piecewise((2**(X2 - n - 2)*(3/2)**(1 - X2)/6, '
+        'X2 - n <= 2), (0, True)), (X2, 1, oo), (n, 1, oo))')
+    assert P(Eq(X1 + X2, 3)) == Rational(1, 12)

rl/Lib/site-packages/sympy/stats/tests/test_error_prop.py

@@ -0,0 +1,60 @@
+from sympy.core.function import Function
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import exp
+from sympy.stats.error_prop import variance_prop
+from sympy.stats.symbolic_probability import (RandomSymbol, Variance,
+        Covariance)
+
+
+def test_variance_prop():
+    x, y, z = symbols('x y z')
+    phi, t = consts = symbols('phi t')
+    a = RandomSymbol(x)
+    var_x = Variance(a)
+    var_y = Variance(RandomSymbol(y))
+    var_z = Variance(RandomSymbol(z))
+    f = Function('f')(x)
+    cases = {
+        x + y: var_x + var_y,
+        a + y: var_x + var_y,
+        x + y + z: var_x + var_y + var_z,
+        2*x: 4*var_x,
+        x*y: var_x*y**2 + var_y*x**2,
+        1/x: var_x/x**4,
+        x/y: (var_x*y**2 + var_y*x**2)/y**4,
+        exp(x): var_x*exp(2*x),
+        exp(2*x): 4*var_x*exp(4*x),
+        exp(-x*t): t**2*var_x*exp(-2*t*x),
+        f: Variance(f),
+        }
+    for inp, out in cases.items():
+        obs = variance_prop(inp, consts=consts)
+        assert out == obs
+
+def test_variance_prop_with_covar():
+    x, y, z = symbols('x y z')
+    phi, t = consts = symbols('phi t')
+    a = RandomSymbol(x)
+    var_x = Variance(a)
+    b = RandomSymbol(y)
+    var_y = Variance(b)
+    c = RandomSymbol(z)
+    var_z = Variance(c)
+    covar_x_y = Covariance(a, b)
+    covar_x_z = Covariance(a, c)
+    covar_y_z = Covariance(b, c)
+    cases = {
+        x + y: var_x + var_y + 2*covar_x_y,
+        a + y: var_x + var_y + 2*covar_x_y,
+        x + y + z: var_x + var_y + var_z + \
+                   2*covar_x_y + 2*covar_x_z + 2*covar_y_z,
+        2*x: 4*var_x,
+        x*y: var_x*y**2 + var_y*x**2 + 2*covar_x_y/(x*y),
+        1/x: var_x/x**4,
+        exp(x): var_x*exp(2*x),
+        exp(2*x): 4*var_x*exp(4*x),
+        exp(-x*t): t**2*var_x*exp(-2*t*x),
+        }
+    for inp, out in cases.items():
+        obs = variance_prop(inp, consts=consts, include_covar=True)
+        assert out == obs

rl/Lib/site-packages/sympy/stats/tests/test_finite_rv.py

@@ -0,0 +1,509 @@
+from sympy.concrete.summations import Sum
+from sympy.core.containers import (Dict, Tuple)
+from sympy.core.function import Function
+from sympy.core.numbers import (I, Rational, nan)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import binomial
+from sympy.functions.combinatorial.numbers import harmonic
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import cos
+from sympy.functions.special.beta_functions import beta
+from sympy.logic.boolalg import (And, Or)
+from sympy.polys.polytools import cancel
+from sympy.sets.sets import FiniteSet
+from sympy.simplify.simplify import simplify
+from sympy.matrices import Matrix
+from sympy.stats import (DiscreteUniform, Die, Bernoulli, Coin, Binomial, BetaBinomial,
+                         Hypergeometric, Rademacher, IdealSoliton, RobustSoliton, P, E, variance,
+                         covariance, skewness, density, where, FiniteRV, pspace, cdf,
+                         correlation, moment, cmoment, smoment, characteristic_function,
+                         moment_generating_function, quantile,  kurtosis, median, coskewness)
+from sympy.stats.frv_types import DieDistribution, BinomialDistribution, \
+    HypergeometricDistribution
+from sympy.stats.rv import Density
+from sympy.testing.pytest import raises
+
+
+def BayesTest(A, B):
+    assert P(A, B) == P(And(A, B)) / P(B)
+    assert P(A, B) == P(B, A) * P(A) / P(B)
+
+
+def test_discreteuniform():
+    # Symbolic
+    a, b, c, t = symbols('a b c t')
+    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)) == S('1/3')
+
+    Y = DiscreteUniform('Y', range(-5, 5))
+
+    # Numeric
+    assert E(Y) == S('-1/2')
+    assert variance(Y) == S('33/4')
+    assert median(Y) == FiniteSet(-1, 0)
+
+    for x in range(-5, 5):
+        assert P(Eq(Y, x)) == S('1/10')
+        assert P(Y <= x) == S(x + 6)/10
+        assert P(Y >= x) == S(5 - x)/10
+
+    assert dict(density(Die('D', 6)).items()) == \
+           dict(density(DiscreteUniform('U', range(1, 7))).items())
+
+    assert characteristic_function(X)(t) == exp(I*a*t)/3 + exp(I*b*t)/3 + exp(I*c*t)/3
+    assert moment_generating_function(X)(t) == exp(a*t)/3 + exp(b*t)/3 + exp(c*t)/3
+    # issue 18611
+    raises(ValueError, lambda: DiscreteUniform('Z', [a, a, a, b, b, c]))
+
+def test_dice():
+    # TODO: Make iid method!
+    X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
+    a, b, t, p = symbols('a b t p')
+
+    assert E(X) == 3 + S.Half
+    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) is S.Zero
+    assert covariance(X, X + Y) == variance(X)
+    assert density(Eq(cos(X*S.Pi), 1))[True] == S.Half
+    assert correlation(X, Y) == 0
+    assert correlation(X, Y) == correlation(Y, X)
+    assert smoment(X + Y, 3) == skewness(X + Y)
+    assert smoment(X + Y, 4) == kurtosis(X + Y)
+    assert smoment(X, 0) == 1
+    assert P(X > 3) == S.Half
+    assert P(2*X > 6) == S.Half
+    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 quantile(X)(p) == Piecewise((nan, (p > 1) | (p < 0)),\
+        (S.One, p <= Rational(1, 6)), (S(2), p <= Rational(1, 3)), (S(3), p <= S.Half),\
+        (S(4), p <= Rational(2, 3)), (S(5), p <= Rational(5, 6)), (S(6), p <= 1))
+
+    assert P(X > 3, X > 3) is S.One
+    assert P(X > Y, Eq(Y, 6)) is S.Zero
+    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[S(22)] == Rational(1, 108) and d[S(4100)] == Rational(1, 216) and S(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)
+
+    assert characteristic_function(X)(t) == exp(6*I*t)/6 + exp(5*I*t)/6 + exp(4*I*t)/6 + exp(3*I*t)/6 + exp(2*I*t)/6 + exp(I*t)/6
+    assert moment_generating_function(X)(t) == exp(6*t)/6 + exp(5*t)/6 + exp(4*t)/6 + exp(3*t)/6 + exp(2*t)/6 + exp(t)/6
+    assert median(X) == FiniteSet(3, 4)
+    D = Die('D', 7)
+    assert median(D) == FiniteSet(4)
+    # Bayes test for die
+    BayesTest(X > 3, X + Y < 5)
+    BayesTest(Eq(X - Y, Z), Z > Y)
+    BayesTest(X > 3, X > 2)
+
+    # arg test for die
+    raises(ValueError, lambda: Die('X', -1))  # issue 8105: negative sides.
+    raises(ValueError, lambda: Die('X', 0))
+    raises(ValueError, lambda: Die('X', 1.5))  # issue 8103: non integer sides.
+
+    # symbolic test for die
+    n, k = symbols('n, k', positive=True)
+    D = Die('D', n)
+    dens = density(D).dict
+    assert dens == Density(DieDistribution(n))
+    assert set(dens.subs(n, 4).doit().keys()) == {1, 2, 3, 4}
+    assert set(dens.subs(n, 4).doit().values()) == {Rational(1, 4)}
+    k = Dummy('k', integer=True)
+    assert E(D).dummy_eq(
+        Sum(Piecewise((k/n, k <= n), (0, True)), (k, 1, n)))
+    assert variance(D).subs(n, 6).doit() == Rational(35, 12)
+
+    ki = Dummy('ki')
+    cumuf = cdf(D)(k)
+    assert cumuf.dummy_eq(
+    Sum(Piecewise((1/n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, k)))
+    assert cumuf.subs({n: 6, k: 2}).doit() == Rational(1, 3)
+
+    t = Dummy('t')
+    cf = characteristic_function(D)(t)
+    assert cf.dummy_eq(
+    Sum(Piecewise((exp(ki*I*t)/n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, n)))
+    assert cf.subs(n, 3).doit() == exp(3*I*t)/3 + exp(2*I*t)/3 + exp(I*t)/3
+    mgf = moment_generating_function(D)(t)
+    assert mgf.dummy_eq(
+    Sum(Piecewise((exp(ki*t)/n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, n)))
+    assert mgf.subs(n, 3).doit() == exp(3*t)/3 + exp(2*t)/3 + exp(t)/3
+
+def test_given():
+    X = Die('X', 6)
+    assert density(X, X > 5) == {S(6): S.One}
+    assert where(X > 2, X > 5).as_boolean() == Eq(X.symbol, 6)
+
+
+def test_domains():
+    X, Y = Die('x', 6), Die('y', 6)
+    x, y = X.symbol, Y.symbol
+    # Domains
+    d = where(X > Y)
+    assert d.condition == (x > y)
+    d = where(And(X > Y, Y > 3))
+    assert d.as_boolean() == Or(And(Eq(x, 5), Eq(y, 4)), And(Eq(x, 6),
+        Eq(y, 5)), And(Eq(x, 6), Eq(y, 4)))
+    assert len(d.elements) == 3
+
+    assert len(pspace(X + Y).domain.elements) == 36
+
+    Z = Die('x', 4)
+
+    raises(ValueError, lambda: P(X > Z))  # Two domains with same internal symbol
+
+    assert pspace(X + Y).domain.set == FiniteSet(1, 2, 3, 4, 5, 6)**2
+
+    assert where(X > 3).set == FiniteSet(4, 5, 6)
+    assert X.pspace.domain.dict == FiniteSet(
+        *[Dict({X.symbol: i}) for i in range(1, 7)])
+
+    assert where(X > Y).dict == FiniteSet(*[Dict({X.symbol: i, Y.symbol: j})
+            for i in range(1, 7) for j in range(1, 7) if i > j])
+
+def test_bernoulli():
+    p, a, b, t = symbols('p a b t')
+    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
+    assert characteristic_function(X)(t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t)
+    assert moment_generating_function(X)(t) == p * exp(a * t) + (-p + 1) * exp(b * t)
+
+    X = Bernoulli('B', p, 1, 0)
+    z = Symbol("z")
+
+    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))
+    assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)), (0, z <= 1 - p), (1, z <= 1))
+    Y = Bernoulli('Y', Rational(1, 2))
+    assert median(Y) == FiniteSet(0, 1)
+    Z = Bernoulli('Z', Rational(2, 3))
+    assert median(Z) == FiniteSet(1)
+    raises(ValueError, lambda: Bernoulli('B', 1.5))
+    raises(ValueError, lambda: Bernoulli('B', -0.5))
+
+    #issue 8248
+    assert X.pspace.compute_expectation(1) == 1
+
+    p = Rational(1, 5)
+    X = Binomial('X', 5, p)
+    Y = Binomial('Y', 7, 2*p)
+    Z = Binomial('Z', 9, 3*p)
+    assert coskewness(Y + Z, X + Y, X + Z).simplify() == 0
+    assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y).simplify() == \
+                        sqrt(1529)*Rational(12, 16819)
+    assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y, X < 2).simplify() \
+                        == -sqrt(357451121)*Rational(2812, 4646864573)
+
+def test_cdf():
+    D = Die('D', 6)
+    o = S.One
+
+    assert cdf(
+        D) == sympify({1: o/6, 2: o/3, 3: o/2, 4: 2*o/3, 5: 5*o/6, 6: o})
+
+
+def test_coins():
+    C, D = Coin('C'), Coin('D')
+    H, T = symbols('H, T')
+    assert P(Eq(C, D)) == S.Half
+    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: S.Half, T: S.Half}
+
+    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))
+
+    raises(ValueError, lambda: P(C > D))  # Can't intelligently compare H to T
+
+def test_binomial_verify_parameters():
+    raises(ValueError, lambda: Binomial('b', .2, .5))
+    raises(ValueError, lambda: Binomial('b', 3, 1.5))
+
+def test_binomial_numeric():
+    nvals = range(5)
+    pvals = [0, Rational(1, 4), S.Half, Rational(3, 4), 1]
+
+    for n in nvals:
+        for p in pvals:
+            X = Binomial('X', n, p)
+            assert E(X) == n*p
+            assert variance(X) == n*p*(1 - p)
+            if n > 0 and 0 < p < 1:
+                assert skewness(X) == (1 - 2*p)/sqrt(n*p*(1 - p))
+                assert kurtosis(X) == 3 + (1 - 6*p*(1 - p))/(n*p*(1 - p))
+            for k in range(n + 1):
+                assert P(Eq(X, k)) == binomial(n, k)*p**k*(1 - p)**(n - k)
+
+def test_binomial_quantile():
+    X = Binomial('X', 50, S.Half)
+    assert quantile(X)(0.95) == S(31)
+    assert median(X) == FiniteSet(25)
+
+    X = Binomial('X', 5, S.Half)
+    p = Symbol("p", positive=True)
+    assert quantile(X)(p) == Piecewise((nan, p > S.One), (S.Zero, p <= Rational(1, 32)),\
+        (S.One, p <= Rational(3, 16)), (S(2), p <= S.Half), (S(3), p <= Rational(13, 16)),\
+        (S(4), p <= Rational(31, 32)), (S(5), p <= S.One))
+    assert median(X) == FiniteSet(2, 3)
+
+
+def test_binomial_symbolic():
+    n = 2
+    p = symbols('p', positive=True)
+    X = Binomial('X', n, p)
+    t = Symbol('t')
+
+    assert simplify(E(X)) == n*p == simplify(moment(X, 1))
+    assert simplify(variance(X)) == n*p*(1 - p) == simplify(cmoment(X, 2))
+    assert cancel(skewness(X) - (1 - 2*p)/sqrt(n*p*(1 - p))) == 0
+    assert cancel((kurtosis(X)) - (3 + (1 - 6*p*(1 - p))/(n*p*(1 - p)))) == 0
+    assert characteristic_function(X)(t) == p ** 2 * exp(2 * I * t) + 2 * p * (-p + 1) * exp(I * t) + (-p + 1) ** 2
+    assert moment_generating_function(X)(t) == p ** 2 * exp(2 * t) + 2 * p * (-p + 1) * exp(t) + (-p + 1) ** 2
+
+    # Test ability to change success/failure winnings
+    H, T = symbols('H T')
+    Y = Binomial('Y', n, p, succ=H, fail=T)
+    assert simplify(E(Y) - (n*(H*p + T*(1 - p)))) == 0
+
+    # test symbolic dimensions
+    n = symbols('n')
+    B = Binomial('B', n, p)
+    raises(NotImplementedError, lambda: P(B > 2))
+    assert density(B).dict == Density(BinomialDistribution(n, p, 1, 0))
+    assert set(density(B).dict.subs(n, 4).doit().keys()) == \
+    {S.Zero, S.One, S(2), S(3), S(4)}
+    assert set(density(B).dict.subs(n, 4).doit().values()) == \
+    {(1 - p)**4, 4*p*(1 - p)**3, 6*p**2*(1 - p)**2, 4*p**3*(1 - p), p**4}
+    k = Dummy('k', integer=True)
+    assert E(B > 2).dummy_eq(
+        Sum(Piecewise((k*p**k*(1 - p)**(-k + n)*binomial(n, k), (k >= 0)
+        & (k <= n) & (k > 2)), (0, True)), (k, 0, n)))
+
+def test_beta_binomial():
+    # verify parameters
+    raises(ValueError, lambda: BetaBinomial('b', .2, 1, 2))
+    raises(ValueError, lambda: BetaBinomial('b', 2, -1, 2))
+    raises(ValueError, lambda: BetaBinomial('b', 2, 1, -2))
+    assert BetaBinomial('b', 2, 1, 1)
+
+    # test numeric values
+    nvals = range(1,5)
+    alphavals = [Rational(1, 4), S.Half, Rational(3, 4), 1, 10]
+    betavals = [Rational(1, 4), S.Half, Rational(3, 4), 1, 10]
+
+    for n in nvals:
+        for a in alphavals:
+            for b in betavals:
+                X = BetaBinomial('X', n, a, b)
+                assert E(X) == moment(X, 1)
+                assert variance(X) == cmoment(X, 2)
+
+    # test symbolic
+    n, a, b = symbols('a b n')
+    assert BetaBinomial('x', n, a, b)
+    n = 2 # Because we're using for loops, can't do symbolic n
+    a, b = symbols('a b', positive=True)
+    X = BetaBinomial('X', n, a, b)
+    t = Symbol('t')
+
+    assert E(X).expand() == moment(X, 1).expand()
+    assert variance(X).expand() == cmoment(X, 2).expand()
+    assert skewness(X) == smoment(X, 3)
+    assert characteristic_function(X)(t) == exp(2*I*t)*beta(a + 2, b)/beta(a, b) +\
+         2*exp(I*t)*beta(a + 1, b + 1)/beta(a, b) + beta(a, b + 2)/beta(a, b)
+    assert moment_generating_function(X)(t) == exp(2*t)*beta(a + 2, b)/beta(a, b) +\
+         2*exp(t)*beta(a + 1, b + 1)/beta(a, b) + beta(a, b + 2)/beta(a, b)
+
+def test_hypergeometric_numeric():
+    for N in range(1, 5):
+        for m in range(0, 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_symbolic():
+    N, m, n = symbols('N, m, n')
+    H = Hypergeometric('H', N, m, n)
+    dens = density(H).dict
+    expec = E(H > 2)
+    assert dens == Density(HypergeometricDistribution(N, m, n))
+    assert dens.subs(N, 5).doit() == Density(HypergeometricDistribution(5, m, n))
+    assert set(dens.subs({N: 3, m: 2, n: 1}).doit().keys()) == {S.Zero, S.One}
+    assert set(dens.subs({N: 3, m: 2, n: 1}).doit().values()) == {Rational(1, 3), Rational(2, 3)}
+    k = Dummy('k', integer=True)
+    assert expec.dummy_eq(
+        Sum(Piecewise((k*binomial(m, k)*binomial(N - m, -k + n)
+        /binomial(N, n), k > 2), (0, True)), (k, 0, n)))
+
+def test_rademacher():
+    X = Rademacher('X')
+    t = Symbol('t')
+
+    assert E(X) == 0
+    assert variance(X) == 1
+    assert density(X)[-1] == S.Half
+    assert density(X)[1] == S.Half
+    assert characteristic_function(X)(t) == exp(I*t)/2 + exp(-I*t)/2
+    assert moment_generating_function(X)(t) == exp(t) / 2 + exp(-t) / 2
+
+def test_ideal_soliton():
+    raises(ValueError, lambda : IdealSoliton('sol', -12))
+    raises(ValueError, lambda : IdealSoliton('sol', 13.2))
+    raises(ValueError, lambda : IdealSoliton('sol', 0))
+    f = Function('f')
+    raises(ValueError, lambda : density(IdealSoliton('sol', 10)).pmf(f))
+
+    k = Symbol('k', integer=True, positive=True)
+    x = Symbol('x', integer=True, positive=True)
+    t = Symbol('t')
+    sol = IdealSoliton('sol', k)
+    assert density(sol).low == S.One
+    assert density(sol).high == k
+    assert density(sol).dict == Density(density(sol))
+    assert density(sol).pmf(x) == Piecewise((1/k, Eq(x, 1)), (1/(x*(x - 1)), k >= x), (0, True))
+
+    k_vals = [5, 20, 50, 100, 1000]
+    for i in k_vals:
+        assert E(sol.subs(k, i)) == harmonic(i) == moment(sol.subs(k, i), 1)
+        assert variance(sol.subs(k, i)) == (i - 1) + harmonic(i) - harmonic(i)**2 == cmoment(sol.subs(k, i),2)
+        assert skewness(sol.subs(k, i)) == smoment(sol.subs(k, i), 3)
+        assert kurtosis(sol.subs(k, i)) == smoment(sol.subs(k, i), 4)
+
+    assert exp(I*t)/10 + Sum(exp(I*t*x)/(x*x - x), (x, 2, k)).subs(k, 10).doit() == characteristic_function(sol.subs(k, 10))(t)
+    assert exp(t)/10 + Sum(exp(t*x)/(x*x - x), (x, 2, k)).subs(k, 10).doit() == moment_generating_function(sol.subs(k, 10))(t)
+
+def test_robust_soliton():
+    raises(ValueError, lambda : RobustSoliton('robSol', -12, 0.1, 0.02))
+    raises(ValueError, lambda : RobustSoliton('robSol', 13, 1.89, 0.1))
+    raises(ValueError, lambda : RobustSoliton('robSol', 15, 0.6, -2.31))
+    f = Function('f')
+    raises(ValueError, lambda : density(RobustSoliton('robSol', 15, 0.6, 0.1)).pmf(f))
+
+    k = Symbol('k', integer=True, positive=True)
+    delta = Symbol('delta', positive=True)
+    c = Symbol('c', positive=True)
+    robSol = RobustSoliton('robSol', k, delta, c)
+    assert density(robSol).low == 1
+    assert density(robSol).high == k
+
+    k_vals = [10, 20, 50]
+    delta_vals = [0.2, 0.4, 0.6]
+    c_vals = [0.01, 0.03, 0.05]
+    for x in k_vals:
+        for y in delta_vals:
+            for z in c_vals:
+                assert E(robSol.subs({k: x, delta: y, c: z})) == moment(robSol.subs({k: x, delta: y, c: z}), 1)
+                assert variance(robSol.subs({k: x, delta: y, c: z})) == cmoment(robSol.subs({k: x, delta: y, c: z}), 2)
+                assert skewness(robSol.subs({k: x, delta: y, c: z})) == smoment(robSol.subs({k: x, delta: y, c: z}), 3)
+                assert kurtosis(robSol.subs({k: x, delta: y, c: z})) == smoment(robSol.subs({k: x, delta: y, c: z}), 4)
+
+def test_FiniteRV():
+    F = FiniteRV('F', {1: S.Half, 2: Rational(1, 4), 3: Rational(1, 4)}, check=True)
+    p = Symbol("p", positive=True)
+
+    assert dict(density(F).items()) == {S.One: S.Half, S(2): Rational(1, 4), S(3): Rational(1, 4)}
+    assert P(F >= 2) == S.Half
+    assert quantile(F)(p) == Piecewise((nan, p > S.One), (S.One, p <= S.Half),\
+        (S(2), p <= Rational(3, 4)),(S(3), True))
+
+    assert pspace(F).domain.as_boolean() == Or(
+        *[Eq(F.symbol, i) for i in [1, 2, 3]])
+
+    assert F.pspace.domain.set == FiniteSet(1, 2, 3)
+    raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S.Half, 3: S.Half}, check=True))
+    raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: Rational(-1, 2), 3: S.One}, check=True))
+    raises(ValueError, lambda: FiniteRV('F', {1: S.One, 2: Rational(3, 2), 3: S.Zero,\
+        4: Rational(-1, 2), 5: Rational(-3, 4), 6: Rational(-1, 4)}, check=True))
+
+    # purposeful invalid pmf but it should not raise since check=False
+    # see test_drv_types.test_ContinuousRV for explanation
+    X = FiniteRV('X', {1: 1, 2: 2})
+    assert E(X) == 5
+    assert P(X <= 2) + P(X > 2) != 1
+
+def test_density_call():
+    from sympy.abc import p
+    x = Bernoulli('x', p)
+    d = density(x)
+    assert d(0) == 1 - p
+    assert d(S.Zero) == 1 - p
+    assert d(5) == 0
+
+    assert 0 in d
+    assert 5 not in d
+    assert d(S.Zero) == d[S.Zero]
+
+
+def test_DieDistribution():
+    from sympy.abc import x
+    X = DieDistribution(6)
+    assert X.pmf(S.Half) is S.Zero
+    assert X.pmf(x).subs({x: 1}).doit() == Rational(1, 6)
+    assert X.pmf(x).subs({x: 7}).doit() == 0
+    assert X.pmf(x).subs({x: -1}).doit() == 0
+    assert X.pmf(x).subs({x: Rational(1, 3)}).doit() == 0
+    raises(ValueError, lambda: X.pmf(Matrix([0, 0])))
+    raises(ValueError, lambda: X.pmf(x**2 - 1))
+
+def test_FinitePSpace():
+    X = Die('X', 6)
+    space = pspace(X)
+    assert space.density == DieDistribution(6)
+
+def test_symbolic_conditions():
+    B = Bernoulli('B', Rational(1, 4))
+    D = Die('D', 4)
+    b, n = symbols('b, n')
+    Y = P(Eq(B, b))
+    Z = E(D > n)
+    assert Y == \
+    Piecewise((Rational(1, 4), Eq(b, 1)), (0, True)) + \
+    Piecewise((Rational(3, 4), Eq(b, 0)), (0, True))
+    assert Z == \
+    Piecewise((Rational(1, 4), n < 1), (0, True)) + Piecewise((S.Half, n < 2), (0, True)) + \
+    Piecewise((Rational(3, 4), n < 3), (0, True)) + Piecewise((S.One, n < 4), (0, True))

rl/Lib/site-packages/sympy/stats/tests/test_joint_rv.py

@@ -0,0 +1,436 @@
+from sympy.concrete.products import Product
+from sympy.concrete.summations import Sum
+from sympy.core.numbers import (Rational, 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 (RisingFactorial, factorial)
+from sympy.functions.elementary.complexes import polar_lift
+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.bessel import besselk
+from sympy.functions.special.gamma_functions import gamma
+from sympy.matrices.dense import eye
+from sympy.matrices.expressions.determinant import Determinant
+from sympy.sets.fancysets import Range
+from sympy.sets.sets import (Interval, ProductSet)
+from sympy.simplify.simplify import simplify
+from sympy.tensor.indexed import (Indexed, IndexedBase)
+from sympy.core.numbers import comp
+from sympy.integrals.integrals import integrate
+from sympy.matrices import Matrix, MatrixSymbol
+from sympy.matrices.expressions.matexpr import MatrixElement
+from sympy.stats import density, median, marginal_distribution, Normal, Laplace, E, sample
+from sympy.stats.joint_rv_types import (JointRV, MultivariateNormalDistribution,
+                JointDistributionHandmade, MultivariateT, NormalGamma,
+                GeneralizedMultivariateLogGammaOmega as GMVLGO, MultivariateBeta,
+                GeneralizedMultivariateLogGamma as GMVLG, MultivariateEwens,
+                Multinomial, NegativeMultinomial, MultivariateNormal,
+                MultivariateLaplace)
+from sympy.testing.pytest import raises, XFAIL, skip, slow
+from sympy.external import import_module
+
+from sympy.abc import x, y
+
+
+
+def test_Normal():
+    m = Normal('A', [1, 2], [[1, 0], [0, 1]])
+    A = MultivariateNormal('A', [1, 2], [[1, 0], [0, 1]])
+    assert m == A
+    assert density(m)(1, 2) == 1/(2*pi)
+    assert m.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
+    raises (ValueError, lambda:m[2])
+    n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+    p = Normal('C',  Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
+    assert density(m)(x, y) == density(p)(x, y)
+    assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
+    raises(ValueError, lambda: marginal_distribution(m))
+    assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1.0
+    N = Normal('N', [1, 2], [[x, 0], [0, y]])
+    assert density(N)(0, 0) == exp(-((4*x + y)/(2*x*y)))/(2*pi*sqrt(x*y))
+
+    raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
+    # symbolic
+    n = symbols('n', integer=True, positive=True)
+    mu = MatrixSymbol('mu', n, 1)
+    sigma = MatrixSymbol('sigma', n, n)
+    X = Normal('X', mu, sigma)
+    assert density(X) == MultivariateNormalDistribution(mu, sigma)
+    raises (NotImplementedError, lambda: median(m))
+    # Below tests should work after issue #17267 is resolved
+    # assert E(X) == mu
+    # assert variance(X) == sigma
+
+    # test symbolic multivariate normal densities
+    n = 3
+
+    Sg = MatrixSymbol('Sg', n, n)
+    mu = MatrixSymbol('mu', n, 1)
+    obs = MatrixSymbol('obs', n, 1)
+
+    X = MultivariateNormal('X', mu, Sg)
+    density_X = density(X)
+
+    eval_a = density_X(obs).subs({Sg: eye(3),
+        mu: Matrix([0, 0, 0]), obs: Matrix([0, 0, 0])}).doit()
+    eval_b = density_X(0, 0, 0).subs({Sg: eye(3), mu: Matrix([0, 0, 0])}).doit()
+
+    assert eval_a == sqrt(2)/(4*pi**Rational(3/2))
+    assert eval_b == sqrt(2)/(4*pi**Rational(3/2))
+
+    n = symbols('n', integer=True, positive=True)
+
+    Sg = MatrixSymbol('Sg', n, n)
+    mu = MatrixSymbol('mu', n, 1)
+    obs = MatrixSymbol('obs', n, 1)
+
+    X = MultivariateNormal('X', mu, Sg)
+    density_X_at_obs = density(X)(obs)
+
+    expected_density = MatrixElement(
+        exp((S(1)/2) * (mu.T - obs.T) * Sg**(-1) * (-mu + obs)) / \
+        sqrt((2*pi)**n * Determinant(Sg)), 0, 0)
+
+    assert density_X_at_obs == expected_density
+
+
+def test_MultivariateTDist():
+    t1 = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
+    assert(density(t1))(1, 1) == 1/(8*pi)
+    assert t1.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
+    assert integrate(density(t1)(x, y), (x, -oo, oo), \
+        (y, -oo, oo)).evalf() == 1.0
+    raises(ValueError, lambda: MultivariateT('T', [1, 2], [[1, 1], [1, -1]], 1))
+    t2 = MultivariateT('t2', [1, 2], [[x, 0], [0, y]], 1)
+    assert density(t2)(1, 2) == 1/(2*pi*sqrt(x*y))
+
+
+def test_multivariate_laplace():
+    raises(ValueError, lambda: Laplace('T', [1, 2], [[1, 2], [2, 1]]))
+    L = Laplace('L', [1, 0], [[1, 0], [0, 1]])
+    L2 = MultivariateLaplace('L2', [1, 0], [[1, 0], [0, 1]])
+    assert density(L)(2, 3) == exp(2)*besselk(0, sqrt(39))/pi
+    L1 = Laplace('L1', [1, 2], [[x, 0], [0, y]])
+    assert density(L1)(0, 1) == \
+        exp(2/y)*besselk(0, sqrt((2 + 4/y + 1/x)/y))/(pi*sqrt(x*y))
+    assert L.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
+    assert L.pspace.distribution == L2.pspace.distribution
+
+
+def test_NormalGamma():
+    ng = NormalGamma('G', 1, 2, 3, 4)
+    assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi)
+    assert ng.pspace.distribution.set == ProductSet(S.Reals, Interval(0, oo))
+    raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1))
+    assert marginal_distribution(ng, 0)(1) == \
+        3*sqrt(10)*gamma(Rational(7, 4))/(10*sqrt(pi)*gamma(Rational(5, 4)))
+    assert marginal_distribution(ng, y)(1) == exp(Rational(-1, 4))/128
+    assert marginal_distribution(ng,[0,1])(x) == x**2*exp(-x/4)/128
+
+
+def test_GeneralizedMultivariateLogGammaDistribution():
+    h = S.Half
+    omega = Matrix([[1, h, h, h],
+                     [h, 1, h, h],
+                     [h, h, 1, h],
+                     [h, h, h, 1]])
+    v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4])
+    y_1, y_2, y_3, y_4 = symbols('y_1:5', real=True)
+    delta = symbols('d', positive=True)
+    G = GMVLGO('G', omega, v, l, mu)
+    Gd = GMVLG('Gd', delta, v, l, mu)
+    dend = ("d**4*Sum(4*24**(-n - 4)*(1 - d)**n*exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 "
+            "+ 4*y_4) - exp(y_1) - exp(2*y_2)/2 - exp(3*y_3)/3 - exp(4*y_4)/4)/"
+            "(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))")
+    assert str(density(Gd)(y_1, y_2, y_3, y_4)) == dend
+    den = ("5*2**(2/3)*5**(1/3)*Sum(4*24**(-n - 4)*(-2**(2/3)*5**(1/3)/4 + 1)**n*"
+          "exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 + 4*y_4) - exp(y_1) - exp(2*y_2)/2 - "
+          "exp(3*y_3)/3 - exp(4*y_4)/4)/(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))/64")
+    assert str(density(G)(y_1, y_2, y_3, y_4)) == den
+    marg = ("5*2**(2/3)*5**(1/3)*exp(4*y_1)*exp(-exp(y_1))*Integral(exp(-exp(4*G[3])"
+            "/4)*exp(16*G[3])*Integral(exp(-exp(3*G[2])/3)*exp(12*G[2])*Integral(exp("
+            "-exp(2*G[1])/2)*exp(8*G[1])*Sum((-1/4)**n*(-4 + 2**(2/3)*5**(1/3"
+            "))**n*exp(n*y_1)*exp(2*n*G[1])*exp(3*n*G[2])*exp(4*n*G[3])/(24**n*gamma(n + 1)"
+            "*gamma(n + 4)**3), (n, 0, oo)), (G[1], -oo, oo)), (G[2], -oo, oo)), (G[3]"
+            ", -oo, oo))/5308416")
+    assert str(marginal_distribution(G, G[0])(y_1)) == marg
+    omega_f1 = Matrix([[1, h, h]])
+    omega_f2 = Matrix([[1, h, h, h],
+                     [h, 1, 2, h],
+                     [h, h, 1, h],
+                     [h, h, h, 1]])
+    omega_f3 = Matrix([[6, h, h, h],
+                     [h, 1, 2, h],
+                     [h, h, 1, h],
+                     [h, h, h, 1]])
+    v_f = symbols("v_f", positive=False, real=True)
+    l_f = [1, 2, v_f, 4]
+    m_f = [v_f, 2, 3, 4]
+    omega_f4 = Matrix([[1, h, h, h, h],
+                     [h, 1, h, h, h],
+                     [h, h, 1, h, h],
+                     [h, h, h, 1, h],
+                     [h, h, h, h, 1]])
+    l_f1 = [1, 2, 3, 4, 5]
+    omega_f5 = Matrix([[1]])
+    mu_f5 = l_f5 = [1]
+
+    raises(ValueError, lambda: GMVLGO('G', omega_f1, v, l, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega_f2, v, l, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega_f3, v, l, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega, v_f, l, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega, v, l_f, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega, v, l, m_f))
+    raises(ValueError, lambda: GMVLGO('G', omega_f4, v, l, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega, v, l_f1, mu))
+    raises(ValueError, lambda: GMVLGO('G', omega_f5, v, l_f5, mu_f5))
+    raises(ValueError, lambda: GMVLG('G', Rational(3, 2), v, l, mu))
+
+
+def test_MultivariateBeta():
+    a1, a2 = symbols('a1, a2', positive=True)
+    a1_f, a2_f = symbols('a1, a2', positive=False, real=True)
+    mb = MultivariateBeta('B', [a1, a2])
+    mb_c = MultivariateBeta('C', a1, a2)
+    assert density(mb)(1, 2) == S(2)**(a2 - 1)*gamma(a1 + a2)/\
+                                (gamma(a1)*gamma(a2))
+    assert marginal_distribution(mb_c, 0)(3) == S(3)**(a1 - 1)*gamma(a1 + a2)/\
+                                                (a2*gamma(a1)*gamma(a2))
+    raises(ValueError, lambda: MultivariateBeta('b1', [a1_f, a2]))
+    raises(ValueError, lambda: MultivariateBeta('b2', [a1, a2_f]))
+    raises(ValueError, lambda: MultivariateBeta('b3', [0, 0]))
+    raises(ValueError, lambda: MultivariateBeta('b4', [a1_f, a2_f]))
+    assert mb.pspace.distribution.set == ProductSet(Interval(0, 1), Interval(0, 1))
+
+
+def test_MultivariateEwens():
+    n, theta, i = symbols('n theta i', positive=True)
+
+    # tests for integer dimensions
+    theta_f = symbols('t_f', negative=True)
+    a = symbols('a_1:4', positive = True, integer = True)
+    ed = MultivariateEwens('E', 3, theta)
+    assert density(ed)(a[0], a[1], a[2]) == Piecewise((6*2**(-a[1])*3**(-a[2])*
+                                            theta**a[0]*theta**a[1]*theta**a[2]/
+                                            (theta*(theta + 1)*(theta + 2)*
+                                            factorial(a[0])*factorial(a[1])*
+                                            factorial(a[2])), Eq(a[0] + 2*a[1] +
+                                            3*a[2], 3)), (0, True))
+    assert marginal_distribution(ed, ed[1])(a[1]) == Piecewise((6*2**(-a[1])*
+                                                    theta**a[1]/((theta + 1)*
+                                                    (theta + 2)*factorial(a[1])),
+                                                    Eq(2*a[1] + 1, 3)), (0, True))
+    raises(ValueError, lambda: MultivariateEwens('e1', 5, theta_f))
+    assert ed.pspace.distribution.set == ProductSet(Range(0, 4, 1),
+                                            Range(0, 2, 1), Range(0, 2, 1))
+
+    # tests for symbolic dimensions
+    eds = MultivariateEwens('E', n, theta)
+    a = IndexedBase('a')
+    j, k = symbols('j, k')
+    den = Piecewise((factorial(n)*Product(theta**a[j]*(j + 1)**(-a[j])/
+           factorial(a[j]), (j, 0, n - 1))/RisingFactorial(theta, n),
+            Eq(n, Sum((k + 1)*a[k], (k, 0, n - 1)))), (0, True))
+    assert density(eds)(a).dummy_eq(den)
+
+
+def test_Multinomial():
+    n, x1, x2, x3, x4 = symbols('n, x1, x2, x3, x4', nonnegative=True, integer=True)
+    p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
+    p1_f, n_f = symbols('p1_f, n_f', negative=True)
+    M = Multinomial('M', n, [p1, p2, p3, p4])
+    C = Multinomial('C', 3, p1, p2, p3)
+    f = factorial
+    assert density(M)(x1, x2, x3, x4) == Piecewise((p1**x1*p2**x2*p3**x3*p4**x4*
+                                            f(n)/(f(x1)*f(x2)*f(x3)*f(x4)),
+                                            Eq(n, x1 + x2 + x3 + x4)), (0, True))
+    assert marginal_distribution(C, C[0])(x1).subs(x1, 1) ==\
+                                                            3*p1*p2**2 +\
+                                                            6*p1*p2*p3 +\
+                                                            3*p1*p3**2
+    raises(ValueError, lambda: Multinomial('b1', 5, [p1, p2, p3, p1_f]))
+    raises(ValueError, lambda: Multinomial('b2', n_f, [p1, p2, p3, p4]))
+    raises(ValueError, lambda: Multinomial('b3', n, 0.5, 0.4, 0.3, 0.1))
+
+
+def test_NegativeMultinomial():
+    k0, x1, x2, x3, x4 = symbols('k0, x1, x2, x3, x4', nonnegative=True, integer=True)
+    p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
+    p1_f = symbols('p1_f', negative=True)
+    N = NegativeMultinomial('N', 4, [p1, p2, p3, p4])
+    C = NegativeMultinomial('C', 4, 0.1, 0.2, 0.3)
+    g = gamma
+    f = factorial
+    assert simplify(density(N)(x1, x2, x3, x4) -
+            p1**x1*p2**x2*p3**x3*p4**x4*(-p1 - p2 - p3 - p4 + 1)**4*g(x1 + x2 +
+            x3 + x4 + 4)/(6*f(x1)*f(x2)*f(x3)*f(x4))) is S.Zero
+    assert comp(marginal_distribution(C, C[0])(1).evalf(), 0.33, .01)
+    raises(ValueError, lambda: NegativeMultinomial('b1', 5, [p1, p2, p3, p1_f]))
+    raises(ValueError, lambda: NegativeMultinomial('b2', k0, 0.5, 0.4, 0.3, 0.4))
+    assert N.pspace.distribution.set == ProductSet(Range(0, oo, 1),
+                    Range(0, oo, 1), Range(0, oo, 1), Range(0, oo, 1))
+
+
+@slow
+def test_JointPSpace_marginal_distribution():
+    T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
+    got = marginal_distribution(T, T[1])(x)
+    ans = sqrt(2)*(x**2/2 + 1)/(4*polar_lift(x**2/2 + 1)**(S(5)/2))
+    assert got == ans, got
+    assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1
+
+    t = MultivariateT('T', [0, 0, 0], [[1, 0, 0], [0, 1, 0], [0, 0, 1]], 3)
+    assert comp(marginal_distribution(t, 0)(1).evalf(), 0.2, .01)
+
+
+def test_JointRV():
+    x1, x2 = (Indexed('x', i) for i in (1, 2))
+    pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi)
+    X = JointRV('x', pdf)
+    assert density(X)(1, 2) == exp(-2)/(2*pi)
+    assert isinstance(X.pspace.distribution, JointDistributionHandmade)
+    assert marginal_distribution(X, 0)(2) == sqrt(2)*exp(Rational(-1, 2))/(2*sqrt(pi))
+
+
+def test_expectation():
+    m = Normal('A', [x, y], [[1, 0], [0, 1]])
+    assert simplify(E(m[1])) == y
+
+
+@XFAIL
+def test_joint_vector_expectation():
+    m = Normal('A', [x, y], [[1, 0], [0, 1]])
+    assert E(m) == (x, y)
+
+
+def test_sample_numpy():
+    distribs_numpy = [
+        MultivariateNormal("M", [3, 4], [[2, 1], [1, 2]]),
+        MultivariateBeta("B", [0.4, 5, 15, 50, 203]),
+        Multinomial("N", 50, [0.3, 0.2, 0.1, 0.25, 0.15])
+    ]
+    size = 3
+    numpy = import_module('numpy')
+    if not numpy:
+        skip('Numpy is not installed. Abort tests for _sample_numpy.')
+    else:
+        for X in distribs_numpy:
+            samps = sample(X, size=size, library='numpy')
+            for sam in samps:
+                assert tuple(sam) in X.pspace.distribution.set
+        N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
+        raises(NotImplementedError, lambda: sample(N_c, library='numpy'))
+
+
+def test_sample_scipy():
+    distribs_scipy = [
+        MultivariateNormal("M", [0, 0], [[0.1, 0.025], [0.025, 0.1]]),
+        MultivariateBeta("B", [0.4, 5, 15]),
+        Multinomial("N", 8, [0.3, 0.2, 0.1, 0.4])
+    ]
+
+    size = 3
+    scipy = import_module('scipy')
+    if not scipy:
+        skip('Scipy not installed. Abort tests for _sample_scipy.')
+    else:
+        for X in distribs_scipy:
+            samps = sample(X, size=size)
+            samps2 = sample(X, size=(2, 2))
+            for sam in samps:
+                assert tuple(sam) in X.pspace.distribution.set
+            for i in range(2):
+                for j in range(2):
+                    assert tuple(samps2[i][j]) in X.pspace.distribution.set
+        N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
+        raises(NotImplementedError, lambda: sample(N_c))
+
+
+def test_sample_pymc():
+    distribs_pymc = [
+        MultivariateNormal("M", [5, 2], [[1, 0], [0, 1]]),
+        MultivariateBeta("B", [0.4, 5, 15]),
+        Multinomial("N", 4, [0.3, 0.2, 0.1, 0.4])
+    ]
+    size = 3
+    pymc = import_module('pymc')
+    if not pymc:
+        skip('PyMC is not installed. Abort tests for _sample_pymc.')
+    else:
+        for X in distribs_pymc:
+            samps = sample(X, size=size, library='pymc')
+            for sam in samps:
+                assert tuple(sam.flatten()) in X.pspace.distribution.set
+        N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
+        raises(NotImplementedError, lambda: sample(N_c, library='pymc'))
+
+
+def test_sample_seed():
+    x1, x2 = (Indexed('x', i) for i in (1, 2))
+    pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi)
+    X = JointRV('x', pdf)
+
+    libraries = ['scipy', 'numpy', 'pymc']
+    for lib in libraries:
+        try:
+            imported_lib = import_module(lib)
+            if imported_lib:
+                s0, s1, s2 = [], [], []
+                s0 = sample(X, size=10, library=lib, seed=0)
+                s1 = sample(X, size=10, library=lib, seed=0)
+                s2 = sample(X, size=10, library=lib, seed=1)
+                assert all(s0 == s1)
+                assert all(s1 != s2)
+        except NotImplementedError:
+            continue
+
+#
+# XXX: This fails for pymc. Previously the test appeared to pass but that is
+# just because the library argument was not passed so the test always used
+# scipy.
+#
+def test_issue_21057():
+    m = Normal("x", [0, 0], [[0, 0], [0, 0]])
+    n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]])
+    p = Normal("x", [0, 0], [[0, 0], [0, 1]])
+    assert m == n
+    libraries = ('scipy', 'numpy')  # , 'pymc')  # <-- pymc fails
+    for library in libraries:
+        try:
+            imported_lib = import_module(library)
+            if imported_lib:
+                s1 = sample(m, size=8, library=library)
+                s2 = sample(n, size=8, library=library)
+                s3 = sample(p, size=8, library=library)
+                assert tuple(s1.flatten()) == tuple(s2.flatten())
+                for s in s3:
+                    assert tuple(s.flatten()) in p.pspace.distribution.set
+        except NotImplementedError:
+            continue
+
+
+#
+# When this passes the pymc part can be uncommented in test_issue_21057 above
+# and this can be deleted.
+#
+@XFAIL
+def test_issue_21057_pymc():
+    m = Normal("x", [0, 0], [[0, 0], [0, 0]])
+    n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]])
+    p = Normal("x", [0, 0], [[0, 0], [0, 1]])
+    assert m == n
+    libraries = ('pymc',)
+    for library in libraries:
+        try:
+            imported_lib = import_module(library)
+            if imported_lib:
+                s1 = sample(m, size=8, library=library)
+                s2 = sample(n, size=8, library=library)
+                s3 = sample(p, size=8, library=library)
+                assert tuple(s1.flatten()) == tuple(s2.flatten())
+                for s in s3:
+                    assert tuple(s.flatten()) in p.pspace.distribution.set
+        except NotImplementedError:
+            continue

rl/Lib/site-packages/sympy/stats/tests/test_matrix_distributions.py

@@ -0,0 +1,186 @@
+from sympy.concrete.products import Product
+from sympy.core.numbers import pi
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, symbols)
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.gamma_functions import gamma
+from sympy.matrices import Determinant, Matrix, Trace, MatrixSymbol, MatrixSet
+from sympy.stats import density, sample
+from sympy.stats.matrix_distributions import (MatrixGammaDistribution,
+                MatrixGamma, MatrixPSpace, Wishart, MatrixNormal, MatrixStudentT)
+from sympy.testing.pytest import raises, skip
+from sympy.external import import_module
+
+
+def test_MatrixPSpace():
+    M = MatrixGammaDistribution(1, 2, [[2, 1], [1, 2]])
+    MP = MatrixPSpace('M', M, 2, 2)
+    assert MP.distribution == M
+    raises(ValueError, lambda: MatrixPSpace('M', M, 1.2, 2))
+
+def test_MatrixGamma():
+    M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]])
+    assert M.pspace.distribution.set == MatrixSet(2, 2, S.Reals)
+    assert isinstance(density(M), MatrixGammaDistribution)
+    X = MatrixSymbol('X', 2, 2)
+    num = exp(Trace(Matrix([[-S(1)/2, 0], [0, -S(1)/2]])*X))
+    assert density(M)(X).doit() == num/(4*pi*sqrt(Determinant(X)))
+    assert density(M)([[2, 1], [1, 2]]).doit() == sqrt(3)*exp(-2)/(12*pi)
+    X = MatrixSymbol('X', 1, 2)
+    Y = MatrixSymbol('Y', 1, 2)
+    assert density(M)([X, Y]).doit() == exp(-X[0, 0]/2 - Y[0, 1]/2)/(4*pi*sqrt(
+                                X[0, 0]*Y[0, 1] - X[0, 1]*Y[0, 0]))
+    # symbolic
+    a, b = symbols('a b', positive=True)
+    d = symbols('d', positive=True, integer=True)
+    Y = MatrixSymbol('Y', d, d)
+    Z = MatrixSymbol('Z', 2, 2)
+    SM = MatrixSymbol('SM', d, d)
+    M2 = MatrixGamma('M2', a, b, SM)
+    M3 = MatrixGamma('M3', 2, 3, [[2, 1], [1, 2]])
+    k = Dummy('k')
+    exprd = pi**(-d*(d - 1)/4)*b**(-a*d)*exp(Trace((-1/b)*SM**(-1)*Y)
+        )*Determinant(SM)**(-a)*Determinant(Y)**(a - d/2 - S(1)/2)/Product(
+        gamma(-k/2 + a + S(1)/2), (k, 1, d))
+    assert density(M2)(Y).dummy_eq(exprd)
+    raises(NotImplementedError, lambda: density(M3 + M)(Z))
+    raises(ValueError, lambda: density(M)(1))
+    raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixGamma('M', -1, -2, [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [2, 1]]))
+    raises(ValueError, lambda: MatrixGamma('M', -1, 2, [[1, 0], [0]]))
+
+def test_Wishart():
+    W = Wishart('W', 5, [[1, 0], [0, 1]])
+    assert W.pspace.distribution.set == MatrixSet(2, 2, S.Reals)
+    X = MatrixSymbol('X', 2, 2)
+    term1 = exp(Trace(Matrix([[-S(1)/2, 0], [0, -S(1)/2]])*X))
+    assert density(W)(X).doit() == term1 * Determinant(X)/(24*pi)
+    assert density(W)([[2, 1], [1, 2]]).doit() == exp(-2)/(8*pi)
+    n = symbols('n', positive=True)
+    d = symbols('d', positive=True, integer=True)
+    Y = MatrixSymbol('Y', d, d)
+    SM = MatrixSymbol('SM', d, d)
+    W = Wishart('W', n, SM)
+    k = Dummy('k')
+    exprd = 2**(-d*n/2)*pi**(-d*(d - 1)/4)*exp(Trace(-(S(1)/2)*SM**(-1)*Y)
+    )*Determinant(SM)**(-n/2)*Determinant(Y)**(
+    -d/2 + n/2 - S(1)/2)/Product(gamma(-k/2 + n/2 + S(1)/2), (k, 1, d))
+    assert density(W)(Y).dummy_eq(exprd)
+    raises(ValueError, lambda: density(W)(1))
+    raises(ValueError, lambda: Wishart('W', -1, [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: Wishart('W', -1, [[1, 0], [2, 1]]))
+    raises(ValueError, lambda: Wishart('W',  2, [[1, 0], [0]]))
+
+def test_MatrixNormal():
+    M = MatrixNormal('M', [[5, 6]], [4], [[2, 1], [1, 2]])
+    assert M.pspace.distribution.set == MatrixSet(1, 2, S.Reals)
+    X = MatrixSymbol('X', 1, 2)
+    term1 = exp(-Trace(Matrix([[ S(2)/3, -S(1)/3], [-S(1)/3, S(2)/3]])*(
+            Matrix([[-5], [-6]]) + X.T)*Matrix([[S(1)/4]])*(Matrix([[-5, -6]]) + X))/2)
+    assert density(M)(X).doit() == (sqrt(3)) * term1/(24*pi)
+    assert density(M)([[7, 8]]).doit() == sqrt(3)*exp(-S(1)/3)/(24*pi)
+    d, n = symbols('d n', positive=True, integer=True)
+    SM2 = MatrixSymbol('SM2', d, d)
+    SM1 = MatrixSymbol('SM1', n, n)
+    LM = MatrixSymbol('LM', n, d)
+    Y = MatrixSymbol('Y', n, d)
+    M = MatrixNormal('M', LM, SM1, SM2)
+    exprd = (2*pi)**(-d*n/2)*exp(-Trace(SM2**(-1)*(-LM.T + Y.T)*SM1**(-1)*(-LM + Y)
+        )/2)*Determinant(SM1)**(-d/2)*Determinant(SM2)**(-n/2)
+    assert density(M)(Y).doit() == exprd
+    raises(ValueError, lambda: density(M)(1))
+    raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [0, 1]], [[1, 0], [2, 1]]))
+    raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2, 1]], [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [0, 1]], [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2]], [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixNormal('M', [1, 2], [[1, 0], [2, 1]], [[1, 0], [0]]))
+    raises(ValueError, lambda: MatrixNormal('M', [[1, 2]], [[1, 0], [0, 1]], [[1, 0]]))
+    raises(ValueError, lambda: MatrixNormal('M', [[1, 2]], [1], [[1, 0]]))
+
+def test_MatrixStudentT():
+    M = MatrixStudentT('M', 2, [[5, 6]], [[2, 1], [1, 2]], [4])
+    assert M.pspace.distribution.set == MatrixSet(1, 2, S.Reals)
+    X = MatrixSymbol('X', 1, 2)
+    D = pi ** (-1.0) * Determinant(Matrix([[4]])) ** (-1.0) * Determinant(Matrix([[2, 1], [1, 2]])) \
+        ** (-0.5) / Determinant(Matrix([[S(1) / 4]]) * (Matrix([[-5, -6]]) + X)
+                                * Matrix([[S(2) / 3, -S(1) / 3], [-S(1) / 3, S(2) / 3]]) * (
+                                        Matrix([[-5], [-6]]) + X.T) + Matrix([[1]])) ** 2
+    assert density(M)(X) == D
+
+    v = symbols('v', positive=True)
+    n, p = 1, 2
+    Omega = MatrixSymbol('Omega', p, p)
+    Sigma = MatrixSymbol('Sigma', n, n)
+    Location = MatrixSymbol('Location', n, p)
+    Y = MatrixSymbol('Y', n, p)
+    M = MatrixStudentT('M', v, Location, Omega, Sigma)
+
+    exprd = gamma(v/2 + 1)*Determinant(Matrix([[1]]) + Sigma**(-1)*(-Location + Y)*Omega**(-1)*(-Location.T + Y.T))**(-v/2 - 1) / \
+            (pi*gamma(v/2)*sqrt(Determinant(Omega))*Determinant(Sigma))
+
+    assert density(M)(Y) == exprd
+    raises(ValueError, lambda: density(M)(1))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [0, 1]], [[1, 0], [2, 1]]))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2, 1]], [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [0, 1]], [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2]], [[1, 0], [0, 1]]))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [1, 2], [[1, 0], [2, 1]], [[1], [2]]))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [[1, 2]], [[1, 0], [0, 1]], [[1, 0]]))
+    raises(ValueError, lambda: MatrixStudentT('M', 1, [[1, 2]], [1], [[1, 0]]))
+    raises(ValueError, lambda: MatrixStudentT('M', -1, [1, 2], [[1, 0], [0, 1]], [4]))
+
+def test_sample_scipy():
+    distribs_scipy = [
+        MatrixNormal('M', [[5, 6]], [4], [[2, 1], [1, 2]]),
+        Wishart('W', 5, [[1, 0], [0, 1]])
+    ]
+
+    size = 5
+    scipy = import_module('scipy')
+    if not scipy:
+        skip('Scipy not installed. Abort tests for _sample_scipy.')
+    else:
+        for X in distribs_scipy:
+            samps = sample(X, size=size)
+            for sam in samps:
+                assert Matrix(sam) in X.pspace.distribution.set
+        M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]])
+        raises(NotImplementedError, lambda: sample(M, size=3))
+
+def test_sample_pymc():
+    distribs_pymc = [
+        MatrixNormal('M', [[5, 6], [3, 4]], [[1, 0], [0, 1]], [[2, 1], [1, 2]]),
+        Wishart('W', 7, [[2, 1], [1, 2]])
+    ]
+    size = 3
+    pymc = import_module('pymc')
+    if not pymc:
+        skip('PyMC is not installed. Abort tests for _sample_pymc.')
+    else:
+        for X in distribs_pymc:
+            samps = sample(X, size=size, library='pymc')
+            for sam in samps:
+                assert Matrix(sam) in X.pspace.distribution.set
+        M = MatrixGamma('M', 1, 2, [[1, 0], [0, 1]])
+        raises(NotImplementedError, lambda: sample(M, size=3))
+
+def test_sample_seed():
+    X = MatrixNormal('M', [[5, 6], [3, 4]], [[1, 0], [0, 1]], [[2, 1], [1, 2]])
+
+    libraries = ['scipy', 'numpy', 'pymc']
+    for lib in libraries:
+        try:
+            imported_lib = import_module(lib)
+            if imported_lib:
+                s0, s1, s2 = [], [], []
+                s0 = sample(X, size=10, library=lib, seed=0)
+                s1 = sample(X, size=10, library=lib, seed=0)
+                s2 = sample(X, size=10, library=lib, seed=1)
+                for i in range(10):
+                    assert (s0[i] == s1[i]).all()
+                    assert (s1[i] != s2[i]).all()
+
+        except NotImplementedError:
+            continue

rl/Lib/site-packages/sympy/stats/tests/test_mix.py

@@ -0,0 +1,82 @@
+from sympy.concrete.summations import Sum
+from sympy.core.add import Add
+from sympy.core.mul import Mul
+from sympy.core.numbers import (Integer, oo, pi)
+from sympy.core.power import Pow
+from sympy.core.relational import (Eq, Ne)
+from sympy.core.symbol import (Dummy, Symbol, 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.delta_functions import DiracDelta
+from sympy.functions.special.gamma_functions import gamma
+from sympy.integrals.integrals import Integral
+from sympy.simplify.simplify import simplify
+from sympy.tensor.indexed import (Indexed, IndexedBase)
+from sympy.functions.elementary.piecewise import ExprCondPair
+from sympy.stats import (Poisson, Beta, Exponential, P,
+                        Multinomial, MultivariateBeta)
+from sympy.stats.crv_types import Normal
+from sympy.stats.drv_types import PoissonDistribution
+from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution
+from sympy.stats.joint_rv import MarginalDistribution
+from sympy.stats.rv import pspace, density
+from sympy.testing.pytest import ignore_warnings
+
+def test_density():
+    x = Symbol('x')
+    l = Symbol('l', positive=True)
+    rate = Beta(l, 2, 3)
+    X = Poisson(x, rate)
+    assert isinstance(pspace(X), CompoundPSpace)
+    assert density(X, Eq(rate, rate.symbol)) == PoissonDistribution(l)
+    N1 = Normal('N1', 0, 1)
+    N2 = Normal('N2', N1, 2)
+    assert density(N2)(0).doit() == sqrt(10)/(10*sqrt(pi))
+    assert simplify(density(N2, Eq(N1, 1))(x)) == \
+        sqrt(2)*exp(-(x - 1)**2/8)/(4*sqrt(pi))
+    assert simplify(density(N2)(x)) == sqrt(10)*exp(-x**2/10)/(10*sqrt(pi))
+
+def test_MarginalDistribution():
+    a1, p1, p2 = symbols('a1 p1 p2', positive=True)
+    C = Multinomial('C', 2, p1, p2)
+    B = MultivariateBeta('B', a1, C[0])
+    MGR = MarginalDistribution(B, (C[0],))
+    mgrc = Mul(Symbol('B'), Piecewise(ExprCondPair(Mul(Integer(2),
+    Pow(Symbol('p1', positive=True), Indexed(IndexedBase(Symbol('C')),
+    Integer(0))), Pow(Symbol('p2', positive=True),
+    Indexed(IndexedBase(Symbol('C')), Integer(1))),
+    Pow(factorial(Indexed(IndexedBase(Symbol('C')), Integer(0))), Integer(-1)),
+    Pow(factorial(Indexed(IndexedBase(Symbol('C')), Integer(1))), Integer(-1))),
+    Eq(Add(Indexed(IndexedBase(Symbol('C')), Integer(0)),
+    Indexed(IndexedBase(Symbol('C')), Integer(1))), Integer(2))),
+    ExprCondPair(Integer(0), True)), Pow(gamma(Symbol('a1', positive=True)),
+    Integer(-1)), gamma(Add(Symbol('a1', positive=True),
+    Indexed(IndexedBase(Symbol('C')), Integer(0)))),
+    Pow(gamma(Indexed(IndexedBase(Symbol('C')), Integer(0))), Integer(-1)),
+    Pow(Indexed(IndexedBase(Symbol('B')), Integer(0)),
+    Add(Symbol('a1', positive=True), Integer(-1))),
+    Pow(Indexed(IndexedBase(Symbol('B')), Integer(1)),
+    Add(Indexed(IndexedBase(Symbol('C')), Integer(0)), Integer(-1))))
+    assert MGR(C) == mgrc
+
+def test_compound_distribution():
+    Y = Poisson('Y', 1)
+    Z = Poisson('Z', Y)
+    assert isinstance(pspace(Z), CompoundPSpace)
+    assert isinstance(pspace(Z).distribution, CompoundDistribution)
+    assert Z.pspace.distribution.pdf(1).doit() == exp(-2)*exp(exp(-1))
+
+def test_mix_expression():
+    Y, E = Poisson('Y', 1), Exponential('E', 1)
+    k = Dummy('k')
+    expr1 = Integral(Sum(exp(-1)*Integral(exp(-k)*DiracDelta(k - 2), (k, 0, oo)
+    )/factorial(k), (k, 0, oo)), (k, -oo, 0))
+    expr2 = Integral(Sum(exp(-1)*Integral(exp(-k)*DiracDelta(k - 2), (k, 0, oo)
+    )/factorial(k), (k, 0, oo)), (k, 0, oo))
+    assert P(Eq(Y + E, 1)) == 0
+    assert P(Ne(Y + E, 2)) == 1
+    with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
+        assert P(E + Y < 2, evaluate=False).rewrite(Integral).dummy_eq(expr1)
+        assert P(E + Y > 2, evaluate=False).rewrite(Integral).dummy_eq(expr2)

rl/Lib/site-packages/sympy/stats/tests/test_random_matrix.py

@@ -0,0 +1,135 @@
+from sympy.concrete.products import Product
+from sympy.core.function import Lambda
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.integrals.integrals import Integral
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.trace import Trace
+from sympy.tensor.indexed import IndexedBase
+from sympy.stats import (GaussianUnitaryEnsemble as GUE, density,
+                         GaussianOrthogonalEnsemble as GOE,
+                         GaussianSymplecticEnsemble as GSE,
+                         joint_eigen_distribution,
+                         CircularUnitaryEnsemble as CUE,
+                         CircularOrthogonalEnsemble as COE,
+                         CircularSymplecticEnsemble as CSE,
+                         JointEigenDistribution,
+                         level_spacing_distribution,
+                         Normal, Beta)
+from sympy.stats.joint_rv_types import JointDistributionHandmade
+from sympy.stats.rv import RandomMatrixSymbol
+from sympy.stats.random_matrix_models import GaussianEnsemble, RandomMatrixPSpace
+from sympy.testing.pytest import raises
+
+def test_GaussianEnsemble():
+    G = GaussianEnsemble('G', 3)
+    assert density(G) == G.pspace.model
+    raises(ValueError, lambda: GaussianEnsemble('G', 3.5))
+
+def test_GaussianUnitaryEnsemble():
+    H = RandomMatrixSymbol('H', 3, 3)
+    G = GUE('U', 3)
+    assert density(G)(H) == sqrt(2)*exp(-3*Trace(H**2)/2)/(4*pi**Rational(9, 2))
+    i, j = (Dummy('i', integer=True, positive=True),
+            Dummy('j', integer=True, positive=True))
+    l = IndexedBase('l')
+    assert joint_eigen_distribution(G).dummy_eq(
+            Lambda((l[1], l[2], l[3]),
+            27*sqrt(6)*exp(-3*(l[1]**2)/2 - 3*(l[2]**2)/2 - 3*(l[3]**2)/2)*
+            Product(Abs(l[i] - l[j])**2, (j, i + 1, 3), (i, 1, 2))/(16*pi**Rational(3, 2))))
+    s = Dummy('s')
+    assert level_spacing_distribution(G).dummy_eq(Lambda(s, 32*s**2*exp(-4*s**2/pi)/pi**2))
+
+
+def test_GaussianOrthogonalEnsemble():
+    H = RandomMatrixSymbol('H', 3, 3)
+    _H = MatrixSymbol('_H', 3, 3)
+    G = GOE('O', 3)
+    assert density(G)(H) == exp(-3*Trace(H**2)/4)/Integral(exp(-3*Trace(_H**2)/4), _H)
+    i, j = (Dummy('i', integer=True, positive=True),
+            Dummy('j', integer=True, positive=True))
+    l = IndexedBase('l')
+    assert joint_eigen_distribution(G).dummy_eq(
+            Lambda((l[1], l[2], l[3]),
+            9*sqrt(2)*exp(-3*l[1]**2/2 - 3*l[2]**2/2 - 3*l[3]**2/2)*
+            Product(Abs(l[i] - l[j]), (j, i + 1, 3), (i, 1, 2))/(32*pi)))
+    s = Dummy('s')
+    assert level_spacing_distribution(G).dummy_eq(Lambda(s, s*pi*exp(-s**2*pi/4)/2))
+
+def test_GaussianSymplecticEnsemble():
+    H = RandomMatrixSymbol('H', 3, 3)
+    _H = MatrixSymbol('_H', 3, 3)
+    G = GSE('O', 3)
+    assert density(G)(H) == exp(-3*Trace(H**2))/Integral(exp(-3*Trace(_H**2)), _H)
+    i, j = (Dummy('i', integer=True, positive=True),
+            Dummy('j', integer=True, positive=True))
+    l = IndexedBase('l')
+    assert joint_eigen_distribution(G).dummy_eq(
+            Lambda((l[1], l[2], l[3]),
+            162*sqrt(3)*exp(-3*l[1]**2/2 - 3*l[2]**2/2 - 3*l[3]**2/2)*
+            Product(Abs(l[i] - l[j])**4, (j, i + 1, 3), (i, 1, 2))/(5*pi**Rational(3, 2))))
+    s = Dummy('s')
+    assert level_spacing_distribution(G).dummy_eq(Lambda(s, S(262144)*s**4*exp(-64*s**2/(9*pi))/(729*pi**3)))
+
+def test_CircularUnitaryEnsemble():
+    CU = CUE('U', 3)
+    j, k = (Dummy('j', integer=True, positive=True),
+            Dummy('k', integer=True, positive=True))
+    t = IndexedBase('t')
+    assert joint_eigen_distribution(CU).dummy_eq(
+            Lambda((t[1], t[2], t[3]),
+            Product(Abs(exp(I*t[j]) - exp(I*t[k]))**2,
+            (j, k + 1, 3), (k, 1, 2))/(48*pi**3))
+    )
+
+def test_CircularOrthogonalEnsemble():
+    CO = COE('U', 3)
+    j, k = (Dummy('j', integer=True, positive=True),
+            Dummy('k', integer=True, positive=True))
+    t = IndexedBase('t')
+    assert joint_eigen_distribution(CO).dummy_eq(
+            Lambda((t[1], t[2], t[3]),
+            Product(Abs(exp(I*t[j]) - exp(I*t[k])),
+            (j, k + 1, 3), (k, 1, 2))/(48*pi**2))
+    )
+
+def test_CircularSymplecticEnsemble():
+    CS = CSE('U', 3)
+    j, k = (Dummy('j', integer=True, positive=True),
+            Dummy('k', integer=True, positive=True))
+    t = IndexedBase('t')
+    assert joint_eigen_distribution(CS).dummy_eq(
+            Lambda((t[1], t[2], t[3]),
+            Product(Abs(exp(I*t[j]) - exp(I*t[k]))**4,
+            (j, k + 1, 3), (k, 1, 2))/(720*pi**3))
+    )
+
+def test_JointEigenDistribution():
+    A = Matrix([[Normal('A00', 0, 1), Normal('A01', 1, 1)],
+                [Beta('A10', 1, 1), Beta('A11', 1, 1)]])
+    assert JointEigenDistribution(A) == \
+    JointDistributionHandmade(-sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)/2 +
+    A[0, 0]/2 + A[1, 1]/2, sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)/2 + A[0, 0]/2 + A[1, 1]/2)
+    raises(ValueError, lambda: JointEigenDistribution(Matrix([[1, 0], [2, 1]])))
+
+def test_issue_19841():
+    G1 = GUE('U', 2)
+    G2 = G1.xreplace({2: 2})
+    assert G1.args == G2.args
+
+    X = MatrixSymbol('X', 2, 2)
+    G = GSE('U', 2)
+    h_pspace = RandomMatrixPSpace('P', model=density(G))
+    H = RandomMatrixSymbol('H', 2, 2, pspace=h_pspace)
+    H2 = RandomMatrixSymbol('H', 2, 2, pspace=None)
+    assert H.doit() == H
+
+    assert (2*H).xreplace({H: X}) == 2*X
+    assert (2*H).xreplace({H2: X}) == 2*H
+    assert (2*H2).xreplace({H: X}) == 2*H2
+    assert (2*H2).xreplace({H2: X}) == 2*X

rl/Lib/site-packages/sympy/stats/tests/test_rv.py

@@ -0,0 +1,441 @@
+from sympy.concrete.summations import Sum
+from sympy.core.basic import Basic
+from sympy.core.containers import Tuple
+from sympy.core.function import Lambda
+from sympy.core.numbers import (Rational, nan, oo, pi)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.combinatorial.factorials import (FallingFactorial, binomial)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.delta_functions import DiracDelta
+from sympy.integrals.integrals import integrate
+from sympy.logic.boolalg import (And, Or)
+from sympy.matrices.dense import Matrix
+from sympy.sets.sets import Interval
+from sympy.tensor.indexed import Indexed
+from sympy.stats import (Die, Normal, Exponential, FiniteRV, P, E, H, variance,
+        density, given, independent, dependent, where, pspace, GaussianUnitaryEnsemble,
+        random_symbols, sample, Geometric, factorial_moment, Binomial, Hypergeometric,
+        DiscreteUniform, Poisson, characteristic_function, moment_generating_function,
+        BernoulliProcess, Variance, Expectation, Probability, Covariance, covariance, cmoment,
+        moment, median)
+from sympy.stats.rv import (IndependentProductPSpace, rs_swap, Density, NamedArgsMixin,
+        RandomSymbol, sample_iter, PSpace, is_random, RandomIndexedSymbol, RandomMatrixSymbol)
+from sympy.testing.pytest import raises, skip, XFAIL, warns_deprecated_sympy
+from sympy.external import import_module
+from sympy.core.numbers import comp
+from sympy.stats.frv_types import BernoulliDistribution
+from sympy.core.symbol import Dummy
+from sympy.functions.elementary.piecewise import Piecewise
+
+def test_where():
+    X, Y = Die('X'), Die('Y')
+    Z = Normal('Z', 0, 1)
+
+    assert where(Z**2 <= 1).set == Interval(-1, 1)
+    assert where(Z**2 <= 1).as_boolean() == Interval(-1, 1).as_relational(Z.symbol)
+    assert where(And(X > Y, Y > 4)).as_boolean() == And(
+        Eq(X.symbol, 6), Eq(Y.symbol, 5))
+
+    assert len(where(X < 3).set) == 2
+    assert 1 in where(X < 3).set
+
+    X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
+    assert where(And(X**2 <= 1, X >= 0)).set == Interval(0, 1)
+    XX = given(X, And(X**2 <= 1, X >= 0))
+    assert XX.pspace.domain.set == Interval(0, 1)
+    assert XX.pspace.domain.as_boolean() == \
+        And(0 <= X.symbol, X.symbol**2 <= 1, -oo < X.symbol, X.symbol < oo)
+
+    with raises(TypeError):
+        XX = given(X, X + 3)
+
+
+def test_random_symbols():
+    X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
+
+    assert set(random_symbols(2*X + 1)) == {X}
+    assert set(random_symbols(2*X + Y)) == {X, Y}
+    assert set(random_symbols(2*X + Y.symbol)) == {X}
+    assert set(random_symbols(2)) == set()
+
+
+def test_characteristic_function():
+    #  Imports I from sympy
+    from sympy.core.numbers import I
+    X = Normal('X',0,1)
+    Y = DiscreteUniform('Y', [1,2,7])
+    Z = Poisson('Z', 2)
+    t = symbols('_t')
+    P = Lambda(t, exp(-t**2/2))
+    Q = Lambda(t, exp(7*t*I)/3 + exp(2*t*I)/3 + exp(t*I)/3)
+    R = Lambda(t, exp(2 * exp(t*I) - 2))
+
+
+    assert characteristic_function(X).dummy_eq(P)
+    assert characteristic_function(Y).dummy_eq(Q)
+    assert characteristic_function(Z).dummy_eq(R)
+
+
+def test_moment_generating_function():
+
+    X = Normal('X',0,1)
+    Y = DiscreteUniform('Y', [1,2,7])
+    Z = Poisson('Z', 2)
+    t = symbols('_t')
+    P = Lambda(t, exp(t**2/2))
+    Q = Lambda(t, (exp(7*t)/3 + exp(2*t)/3 + exp(t)/3))
+    R = Lambda(t, exp(2 * exp(t) - 2))
+
+
+    assert moment_generating_function(X).dummy_eq(P)
+    assert moment_generating_function(Y).dummy_eq(Q)
+    assert moment_generating_function(Z).dummy_eq(R)
+
+def test_sample_iter():
+
+    X = Normal('X',0,1)
+    Y = DiscreteUniform('Y', [1, 2, 7])
+    Z = Poisson('Z', 2)
+
+    scipy = import_module('scipy')
+    if not scipy:
+        skip('Scipy is not installed. Abort tests')
+    expr = X**2 + 3
+    iterator = sample_iter(expr)
+
+    expr2 = Y**2 + 5*Y + 4
+    iterator2 = sample_iter(expr2)
+
+    expr3 = Z**3 + 4
+    iterator3 = sample_iter(expr3)
+
+    def is_iterator(obj):
+        if (
+            hasattr(obj, '__iter__') and
+            (hasattr(obj, 'next') or
+            hasattr(obj, '__next__')) and
+            callable(obj.__iter__) and
+            obj.__iter__() is obj
+           ):
+            return True
+        else:
+            return False
+    assert is_iterator(iterator)
+    assert is_iterator(iterator2)
+    assert is_iterator(iterator3)
+
+def test_pspace():
+    X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
+    x = Symbol('x')
+
+    raises(ValueError, lambda: pspace(5 + 3))
+    raises(ValueError, lambda: pspace(x < 1))
+    assert pspace(X) == X.pspace
+    assert pspace(2*X + 1) == X.pspace
+    assert pspace(2*X + Y) == IndependentProductPSpace(Y.pspace, X.pspace)
+
+def test_rs_swap():
+    X = Normal('x', 0, 1)
+    Y = Exponential('y', 1)
+
+    XX = Normal('x', 0, 2)
+    YY = Normal('y', 0, 3)
+
+    expr = 2*X + Y
+    assert expr.subs(rs_swap((X, Y), (YY, XX))) == 2*XX + YY
+
+
+def test_RandomSymbol():
+
+    X = Normal('x', 0, 1)
+    Y = Normal('x', 0, 2)
+    assert X.symbol == Y.symbol
+    assert X != Y
+
+    assert X.name == X.symbol.name
+
+    X = Normal('lambda', 0, 1) # make sure we can use protected terms
+    X = Normal('Lambda', 0, 1) # make sure we can use SymPy terms
+
+
+def test_RandomSymbol_diff():
+    X = Normal('x', 0, 1)
+    assert (2*X).diff(X)
+
+
+def test_random_symbol_no_pspace():
+    x = RandomSymbol(Symbol('x'))
+    assert x.pspace == PSpace()
+
+def test_overlap():
+    X = Normal('x', 0, 1)
+    Y = Normal('x', 0, 2)
+
+    raises(ValueError, lambda: P(X > Y))
+
+
+def test_IndependentProductPSpace():
+    X = Normal('X', 0, 1)
+    Y = Normal('Y', 0, 1)
+    px = X.pspace
+    py = Y.pspace
+    assert pspace(X + Y) == IndependentProductPSpace(px, py)
+    assert pspace(X + Y) == IndependentProductPSpace(py, px)
+
+
+def test_E():
+    assert E(5) == 5
+
+
+def test_H():
+    X = Normal('X', 0, 1)
+    D = Die('D', sides = 4)
+    G = Geometric('G', 0.5)
+    assert H(X, X > 0) == -log(2)/2 + S.Half + log(pi)/2
+    assert H(D, D > 2) == log(2)
+    assert comp(H(G).evalf().round(2), 1.39)
+
+
+def test_Sample():
+    X = Die('X', 6)
+    Y = Normal('Y', 0, 1)
+    z = Symbol('z', integer=True)
+
+    scipy = import_module('scipy')
+    if not scipy:
+        skip('Scipy is not installed. Abort tests')
+    assert sample(X) in [1, 2, 3, 4, 5, 6]
+    assert isinstance(sample(X + Y), float)
+
+    assert P(X + Y > 0, Y < 0, numsamples=10).is_number
+    assert E(X + Y, numsamples=10).is_number
+    assert E(X**2 + Y, numsamples=10).is_number
+    assert E((X + Y)**2, numsamples=10).is_number
+    assert variance(X + Y, numsamples=10).is_number
+
+    raises(TypeError, lambda: P(Y > z, numsamples=5))
+
+    assert P(sin(Y) <= 1, numsamples=10) == 1.0
+    assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10) == 1.0
+
+    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))
+
+    numpy = import_module('numpy')
+    if not numpy:
+        skip('Numpy is not installed. Abort tests')
+    #Test Issue #21563: Output of sample must be a float or array
+    assert isinstance(sample(X), (numpy.int32, numpy.int64))
+    assert isinstance(sample(Y), numpy.float64)
+    assert isinstance(sample(X, size=2), numpy.ndarray)
+
+    with warns_deprecated_sympy():
+        sample(X, numsamples=2)
+
+@XFAIL
+def test_samplingE():
+    scipy = import_module('scipy')
+    if not scipy:
+        skip('Scipy is not installed. Abort tests')
+    Y = Normal('Y', 0, 1)
+    z = Symbol('z', integer=True)
+    assert E(Sum(1/z**Y, (z, 1, oo)), Y > 2, numsamples=3).is_number
+
+
+def test_given():
+    X = Normal('X', 0, 1)
+    Y = Normal('Y', 0, 1)
+    A = given(X, True)
+    B = given(X, Y > 2)
+
+    assert X == A == B
+
+
+def test_factorial_moment():
+    X = Poisson('X', 2)
+    Y = Binomial('Y', 2, S.Half)
+    Z = Hypergeometric('Z', 4, 2, 2)
+    assert factorial_moment(X, 2) == 4
+    assert factorial_moment(Y, 2) == S.Half
+    assert factorial_moment(Z, 2) == Rational(1, 3)
+
+    x, y, z, l = symbols('x y z l')
+    Y = Binomial('Y', 2, y)
+    Z = Hypergeometric('Z', 10, 2, 3)
+    assert factorial_moment(Y, l) == y**2*FallingFactorial(
+        2, l) + 2*y*(1 - y)*FallingFactorial(1, l) + (1 - y)**2*\
+            FallingFactorial(0, l)
+    assert factorial_moment(Z, l) == 7*FallingFactorial(0, l)/\
+        15 + 7*FallingFactorial(1, l)/15 + FallingFactorial(2, l)/15
+
+
+def test_dependence():
+    X, Y = Die('X'), Die('Y')
+    assert independent(X, 2*Y)
+    assert not dependent(X, 2*Y)
+
+    X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
+    assert independent(X, Y)
+    assert dependent(X, 2*X)
+
+    # Create a dependency
+    XX, YY = given(Tuple(X, Y), Eq(X + Y, 3))
+    assert dependent(XX, YY)
+
+def test_dependent_finite():
+    X, Y = Die('X'), Die('Y')
+    # Dependence testing requires symbolic conditions which currently break
+    # finite random variables
+    assert dependent(X, Y + X)
+
+    XX, YY = given(Tuple(X, Y), X + Y > 5)  # Create a dependency
+    assert dependent(XX, YY)
+
+
+def test_normality():
+    X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
+    x = Symbol('x', real=True)
+    z = Symbol('z', real=True)
+    dens = density(X - Y, Eq(X + Y, z))
+
+    assert integrate(dens(x), (x, -oo, oo)) == 1
+
+
+def test_Density():
+    X = Die('X', 6)
+    d = Density(X)
+    assert d.doit() == density(X)
+
+def test_NamedArgsMixin():
+    class Foo(Basic, NamedArgsMixin):
+        _argnames = 'foo', 'bar'
+
+    a = Foo(S(1), S(2))
+
+    assert a.foo == 1
+    assert a.bar == 2
+
+    raises(AttributeError, lambda: a.baz)
+
+    class Bar(Basic, NamedArgsMixin):
+        pass
+
+    raises(AttributeError, lambda: Bar(S(1), S(2)).foo)
+
+def test_density_constant():
+    assert density(3)(2) == 0
+    assert density(3)(3) == DiracDelta(0)
+
+def test_cmoment_constant():
+    assert variance(3) == 0
+    assert cmoment(3, 3) == 0
+    assert cmoment(3, 4) == 0
+    x = Symbol('x')
+    assert variance(x) == 0
+    assert cmoment(x, 15) == 0
+    assert cmoment(x, 0) == 1
+
+def test_moment_constant():
+    assert moment(3, 0) == 1
+    assert moment(3, 1) == 3
+    assert moment(3, 2) == 9
+    x = Symbol('x')
+    assert moment(x, 2) == x**2
+
+def test_median_constant():
+    assert median(3) == 3
+    x = Symbol('x')
+    assert median(x) == x
+
+def test_real():
+    x = Normal('x', 0, 1)
+    assert x.is_real
+
+
+def test_issue_10052():
+    X = Exponential('X', 3)
+    assert P(X < oo) == 1
+    assert P(X > oo) == 0
+    assert P(X < 2, X > oo) == 0
+    assert P(X < oo, X > oo) == 0
+    assert P(X < oo, X > 2) == 1
+    assert P(X < 3, X == 2) == 0
+    raises(ValueError, lambda: P(1))
+    raises(ValueError, lambda: P(X < 1, 2))
+
+def test_issue_11934():
+    density = {0: .5, 1: .5}
+    X = FiniteRV('X', density)
+    assert E(X) == 0.5
+    assert P( X>= 2) == 0
+
+def test_issue_8129():
+    X = Exponential('X', 4)
+    assert P(X >= X) == 1
+    assert P(X > X) == 0
+    assert P(X > X+1) == 0
+
+def test_issue_12237():
+    X = Normal('X', 0, 1)
+    Y = Normal('Y', 0, 1)
+    U = P(X > 0, X)
+    V = P(Y < 0, X)
+    W = P(X + Y > 0, X)
+    assert W == P(X + Y > 0, X)
+    assert U == BernoulliDistribution(S.Half, S.Zero, S.One)
+    assert V == S.Half
+
+def test_is_random():
+    X = Normal('X', 0, 1)
+    Y = Normal('Y', 0, 1)
+    a, b = symbols('a, b')
+    G = GaussianUnitaryEnsemble('U', 2)
+    B = BernoulliProcess('B', 0.9)
+    assert not is_random(a)
+    assert not is_random(a + b)
+    assert not is_random(a * b)
+    assert not is_random(Matrix([a**2, b**2]))
+    assert is_random(X)
+    assert is_random(X**2 + Y)
+    assert is_random(Y + b**2)
+    assert is_random(Y > 5)
+    assert is_random(B[3] < 1)
+    assert is_random(G)
+    assert is_random(X * Y * B[1])
+    assert is_random(Matrix([[X, B[2]], [G, Y]]))
+    assert is_random(Eq(X, 4))
+
+def test_issue_12283():
+    x = symbols('x')
+    X = RandomSymbol(x)
+    Y = RandomSymbol('Y')
+    Z = RandomMatrixSymbol('Z', 2, 1)
+    W = RandomMatrixSymbol('W', 2, 1)
+    RI = RandomIndexedSymbol(Indexed('RI', 3))
+    assert pspace(Z) == PSpace()
+    assert pspace(RI) == PSpace()
+    assert pspace(X) == PSpace()
+    assert E(X) == Expectation(X)
+    assert P(Y > 3) == Probability(Y > 3)
+    assert variance(X) == Variance(X)
+    assert variance(RI) == Variance(RI)
+    assert covariance(X, Y) == Covariance(X, Y)
+    assert covariance(W, Z) == Covariance(W, Z)
+
+def test_issue_6810():
+    X = Die('X', 6)
+    Y = Normal('Y', 0, 1)
+    assert P(Eq(X, 2)) == S(1)/6
+    assert P(Eq(Y, 0)) == 0
+    assert P(Or(X > 2, X < 3)) == 1
+    assert P(And(X > 3, X > 2)) == S(1)/2
+
+def test_issue_20286():
+    n, p = symbols('n p')
+    B = Binomial('B', n, p)
+    k = Dummy('k', integer = True)
+    eq = Sum(Piecewise((-p**k*(1 - p)**(-k + n)*log(p**k*(1 - p)**(-k + n)*binomial(n, k))*binomial(n, k), (k >= 0) & (k <= n)), (nan, True)), (k, 0, n))
+    assert eq.dummy_eq(H(B))

rl/Lib/site-packages/sympy/stats/tests/test_stochastic_process.py

@@ -0,0 +1,763 @@
+from sympy.concrete.summations import Sum
+from sympy.core.containers import Tuple
+from sympy.core.function import Lambda
+from sympy.core.numbers import (Float, Rational, oo, pi)
+from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.integers import ceiling
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.error_functions import erf
+from sympy.functions.special.gamma_functions import (gamma, lowergamma)
+from sympy.logic.boolalg import (And, Not)
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.immutable import ImmutableMatrix
+from sympy.sets.contains import Contains
+from sympy.sets.fancysets import Range
+from sympy.sets.sets import (FiniteSet, Interval)
+from sympy.stats import (DiscreteMarkovChain, P, TransitionMatrixOf, E,
+                         StochasticStateSpaceOf, variance, ContinuousMarkovChain,
+                         BernoulliProcess, PoissonProcess, WienerProcess,
+                         GammaProcess, sample_stochastic_process)
+from sympy.stats.joint_rv import JointDistribution
+from sympy.stats.joint_rv_types import JointDistributionHandmade
+from sympy.stats.rv import RandomIndexedSymbol
+from sympy.stats.symbolic_probability import Probability, Expectation
+from sympy.testing.pytest import (raises, skip, ignore_warnings,
+                                  warns_deprecated_sympy)
+from sympy.external import import_module
+from sympy.stats.frv_types import BernoulliDistribution
+from sympy.stats.drv_types import PoissonDistribution
+from sympy.stats.crv_types import NormalDistribution, GammaDistribution
+from sympy.core.symbol import Str
+
+
+def test_DiscreteMarkovChain():
+
+    # pass only the name
+    X = DiscreteMarkovChain("X")
+    assert isinstance(X.state_space, Range)
+    assert X.index_set == S.Naturals0
+    assert isinstance(X.transition_probabilities, MatrixSymbol)
+    t = symbols('t', positive=True, integer=True)
+    assert isinstance(X[t], RandomIndexedSymbol)
+    assert E(X[0]) == Expectation(X[0])
+    raises(TypeError, lambda: DiscreteMarkovChain(1))
+    raises(NotImplementedError, lambda: X(t))
+    raises(NotImplementedError, lambda: X.communication_classes())
+    raises(NotImplementedError, lambda: X.canonical_form())
+    raises(NotImplementedError, lambda: X.decompose())
+
+    nz = Symbol('n', integer=True)
+    TZ = MatrixSymbol('M', nz, nz)
+    SZ = Range(nz)
+    YZ = DiscreteMarkovChain('Y', SZ, TZ)
+    assert P(Eq(YZ[2], 1), Eq(YZ[1], 0)) == TZ[0, 1]
+
+    raises(ValueError, lambda: sample_stochastic_process(t))
+    raises(ValueError, lambda: next(sample_stochastic_process(X)))
+    # pass name and state_space
+    # any hashable object should be a valid state
+    # states should be valid as a tuple/set/list/Tuple/Range
+    sym, rainy, cloudy, sunny = symbols('a Rainy Cloudy Sunny', real=True)
+    state_spaces = [(1, 2, 3), [Str('Hello'), sym, DiscreteMarkovChain("Y", (1,2,3))],
+                    Tuple(S(1), exp(sym), Str('World'), sympify=False), Range(-1, 5, 2),
+                    [rainy, cloudy, sunny]]
+    chains = [DiscreteMarkovChain("Y", state_space) for state_space in state_spaces]
+
+    for i, Y in enumerate(chains):
+        assert isinstance(Y.transition_probabilities, MatrixSymbol)
+        assert Y.state_space == state_spaces[i] or Y.state_space == FiniteSet(*state_spaces[i])
+        assert Y.number_of_states == 3
+
+        with ignore_warnings(UserWarning):  # TODO: Restore tests once warnings are removed
+            assert P(Eq(Y[2], 1), Eq(Y[0], 2), evaluate=False) == Probability(Eq(Y[2], 1), Eq(Y[0], 2))
+        assert E(Y[0]) == Expectation(Y[0])
+
+        raises(ValueError, lambda: next(sample_stochastic_process(Y)))
+
+    raises(TypeError, lambda: DiscreteMarkovChain("Y", {1: 1}))
+    Y = DiscreteMarkovChain("Y", Range(1, t, 2))
+    assert Y.number_of_states == ceiling((t-1)/2)
+
+    # pass name and transition_probabilities
+    chains = [DiscreteMarkovChain("Y", trans_probs=Matrix([[]])),
+              DiscreteMarkovChain("Y", trans_probs=Matrix([[0, 1], [1, 0]])),
+              DiscreteMarkovChain("Y", trans_probs=Matrix([[pi, 1-pi], [sym, 1-sym]]))]
+    for Z in chains:
+        assert Z.number_of_states == Z.transition_probabilities.shape[0]
+        assert isinstance(Z.transition_probabilities, ImmutableMatrix)
+
+    # pass name, state_space and transition_probabilities
+    T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
+    TS = MatrixSymbol('T', 3, 3)
+    Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
+    YS = DiscreteMarkovChain("Y", ['One', 'Two', 3], TS)
+    assert Y.joint_distribution(1, Y[2], 3) == JointDistribution(Y[1], Y[2], Y[3])
+    raises(ValueError, lambda: Y.joint_distribution(Y[1].symbol, Y[2].symbol))
+    assert P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2) == Float(0.36, 2)
+    assert (P(Eq(YS[3], 2), Eq(YS[1], 1)) -
+            (TS[0, 2]*TS[1, 0] + TS[1, 1]*TS[1, 2] + TS[1, 2]*TS[2, 2])).simplify() == 0
+    assert P(Eq(YS[1], 1), Eq(YS[2], 2)) == Probability(Eq(YS[1], 1))
+    assert P(Eq(YS[3], 3), Eq(YS[1], 1)) == TS[0, 2]*TS[1, 0] + TS[1, 1]*TS[1, 2] + TS[1, 2]*TS[2, 2]
+    TO = Matrix([[0.25, 0.75, 0],[0, 0.25, 0.75],[0.75, 0, 0.25]])
+    assert P(Eq(Y[3], 2), Eq(Y[1], 1) & TransitionMatrixOf(Y, TO)).round(3) == Float(0.375, 3)
+    with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
+        assert E(Y[3], evaluate=False) == Expectation(Y[3])
+        assert E(Y[3], Eq(Y[2], 1)).round(2) == Float(1.1, 3)
+    TSO = MatrixSymbol('T', 4, 4)
+    raises(ValueError, lambda: str(P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TSO))))
+    raises(TypeError, lambda: DiscreteMarkovChain("Z", [0, 1, 2], symbols('M')))
+    raises(ValueError, lambda: DiscreteMarkovChain("Z", [0, 1, 2], MatrixSymbol('T', 3, 4)))
+    raises(ValueError, lambda: E(Y[3], Eq(Y[2], 6)))
+    raises(ValueError, lambda: E(Y[2], Eq(Y[3], 1)))
+
+
+    # extended tests for probability queries
+    TO1 = Matrix([[Rational(1, 4), Rational(3, 4), 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]])
+    assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)),
+            Eq(Probability(Eq(Y[0], 0)), Rational(1, 4)) & TransitionMatrixOf(Y, TO1)) == Rational(1, 16)
+    assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), TransitionMatrixOf(Y, TO1)) == \
+            Probability(Eq(Y[0], 0))/4
+    assert P(Lt(X[1], 2) & Gt(X[1], 0), Eq(X[0], 2) &
+        StochasticStateSpaceOf(X, [0, 1, 2]) & TransitionMatrixOf(X, TO1)) == Rational(1, 4)
+    assert P(Lt(X[1], 2) & Gt(X[1], 0), Eq(X[0], 2) &
+             StochasticStateSpaceOf(X, [S(0), '0', 1]) & TransitionMatrixOf(X, TO1)) == Rational(1, 4)
+    assert P(Ne(X[1], 2) & Ne(X[1], 1), Eq(X[0], 2) &
+        StochasticStateSpaceOf(X, [0, 1, 2]) & TransitionMatrixOf(X, TO1)) is S.Zero
+    assert P(Ne(X[1], 2) & Ne(X[1], 1), Eq(X[0], 2) &
+             StochasticStateSpaceOf(X, [S(0), '0', 1]) & TransitionMatrixOf(X, TO1)) is S.Zero
+    assert P(And(Eq(Y[2], 1), Eq(Y[1], 1), Eq(Y[0], 0)), Eq(Y[1], 1)) == 0.1*Probability(Eq(Y[0], 0))
+
+    # testing properties of Markov chain
+    TO2 = Matrix([[S.One, 0, 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)],[0, Rational(1, 4), Rational(3, 4)]])
+    TO3 = Matrix([[Rational(1, 4), Rational(3, 4), 0],[Rational(1, 3), Rational(1, 3), Rational(1, 3)], [0, Rational(1, 4), Rational(3, 4)]])
+    Y2 = DiscreteMarkovChain('Y', trans_probs=TO2)
+    Y3 = DiscreteMarkovChain('Y', trans_probs=TO3)
+    assert Y3.fundamental_matrix() == ImmutableMatrix([[176, 81, -132], [36, 141, -52], [-44, -39, 208]])/125
+    assert Y2.is_absorbing_chain() == True
+    assert Y3.is_absorbing_chain() == False
+    assert Y2.canonical_form() == ([0, 1, 2], TO2)
+    assert Y3.canonical_form() == ([0, 1, 2], TO3)
+    assert Y2.decompose() == ([0, 1, 2], TO2[0:1, 0:1], TO2[1:3, 0:1], TO2[1:3, 1:3])
+    assert Y3.decompose() == ([0, 1, 2], TO3, Matrix(0, 3, []), Matrix(0, 0, []))
+    TO4 = Matrix([[Rational(1, 5), Rational(2, 5), Rational(2, 5)], [Rational(1, 10), S.Half, Rational(2, 5)], [Rational(3, 5), Rational(3, 10), Rational(1, 10)]])
+    Y4 = DiscreteMarkovChain('Y', trans_probs=TO4)
+    w = ImmutableMatrix([[Rational(11, 39), Rational(16, 39), Rational(4, 13)]])
+    assert Y4.limiting_distribution == w
+    assert Y4.is_regular() == True
+    assert Y4.is_ergodic() == True
+    TS1 = MatrixSymbol('T', 3, 3)
+    Y5 = DiscreteMarkovChain('Y', trans_probs=TS1)
+    assert Y5.limiting_distribution(w, TO4).doit() == True
+    assert Y5.stationary_distribution(condition_set=True).subs(TS1, TO4).contains(w).doit() == S.true
+    TO6 = Matrix([[S.One, 0, 0, 0, 0],[S.Half, 0, S.Half, 0, 0],[0, S.Half, 0, S.Half, 0], [0, 0, S.Half, 0, S.Half], [0, 0, 0, 0, 1]])
+    Y6 = DiscreteMarkovChain('Y', trans_probs=TO6)
+    assert Y6.fundamental_matrix() == ImmutableMatrix([[Rational(3, 2), S.One, S.Half], [S.One, S(2), S.One], [S.Half, S.One, Rational(3, 2)]])
+    assert Y6.absorbing_probabilities() == ImmutableMatrix([[Rational(3, 4), Rational(1, 4)], [S.Half, S.Half], [Rational(1, 4), Rational(3, 4)]])
+    with warns_deprecated_sympy():
+        Y6.absorbing_probabilites()
+    TO7 = Matrix([[Rational(1, 2), Rational(1, 4), Rational(1, 4)], [Rational(1, 2), 0, Rational(1, 2)], [Rational(1, 4), Rational(1, 4), Rational(1, 2)]])
+    Y7 = DiscreteMarkovChain('Y', trans_probs=TO7)
+    assert Y7.is_absorbing_chain() == False
+    assert Y7.fundamental_matrix() == ImmutableMatrix([[Rational(86, 75), Rational(1, 25), Rational(-14, 75)],
+                                                            [Rational(2, 25), Rational(21, 25), Rational(2, 25)],
+                                                            [Rational(-14, 75), Rational(1, 25), Rational(86, 75)]])
+
+    # test for zero-sized matrix functionality
+    X = DiscreteMarkovChain('X', trans_probs=Matrix([[]]))
+    assert X.number_of_states == 0
+    assert X.stationary_distribution() == Matrix([[]])
+    assert X.communication_classes() == []
+    assert X.canonical_form() == ([], Matrix([[]]))
+    assert X.decompose() == ([], Matrix([[]]), Matrix([[]]), Matrix([[]]))
+    assert X.is_regular() == False
+    assert X.is_ergodic() == False
+
+    # test communication_class
+    # see https://drive.google.com/drive/folders/1HbxLlwwn2b3U8Lj7eb_ASIUb5vYaNIjg?usp=sharing
+    # tutorial 2.pdf
+    TO7 = Matrix([[0, 5, 5, 0, 0],
+                  [0, 0, 0, 10, 0],
+                  [5, 0, 5, 0, 0],
+                  [0, 10, 0, 0, 0],
+                  [0, 3, 0, 3, 4]])/10
+    Y7 = DiscreteMarkovChain('Y', trans_probs=TO7)
+    tuples = Y7.communication_classes()
+    classes, recurrence, periods = list(zip(*tuples))
+    assert classes == ([1, 3], [0, 2], [4])
+    assert recurrence == (True, False, False)
+    assert periods == (2, 1, 1)
+
+    TO8 = Matrix([[0, 0, 0, 10, 0, 0],
+                  [5, 0, 5, 0, 0, 0],
+                  [0, 4, 0, 0, 0, 6],
+                  [10, 0, 0, 0, 0, 0],
+                  [0, 10, 0, 0, 0, 0],
+                  [0, 0, 0, 5, 5, 0]])/10
+    Y8 = DiscreteMarkovChain('Y', trans_probs=TO8)
+    tuples = Y8.communication_classes()
+    classes, recurrence, periods = list(zip(*tuples))
+    assert classes == ([0, 3], [1, 2, 5, 4])
+    assert recurrence == (True, False)
+    assert periods == (2, 2)
+
+    TO9 = Matrix([[2, 0, 0, 3, 0, 0, 3, 2, 0, 0],
+                  [0, 10, 0, 0, 0, 0, 0, 0, 0, 0],
+                  [0, 2, 2, 0, 0, 0, 0, 0, 3, 3],
+                  [0, 0, 0, 3, 0, 0, 6, 1, 0, 0],
+                  [0, 0, 0, 0, 5, 5, 0, 0, 0, 0],
+                  [0, 0, 0, 0, 0, 10, 0, 0, 0, 0],
+                  [4, 0, 0, 5, 0, 0, 1, 0, 0, 0],
+                  [2, 0, 0, 4, 0, 0, 2, 2, 0, 0],
+                  [3, 0, 1, 0, 0, 0, 0, 0, 4, 2],
+                  [0, 0, 4, 0, 0, 0, 0, 0, 3, 3]])/10
+    Y9 = DiscreteMarkovChain('Y', trans_probs=TO9)
+    tuples = Y9.communication_classes()
+    classes, recurrence, periods = list(zip(*tuples))
+    assert classes == ([0, 3, 6, 7], [1], [2, 8, 9], [5], [4])
+    assert recurrence == (True, True, False, True, False)
+    assert periods == (1, 1, 1, 1, 1)
+
+    # test canonical form
+    # see https://web.archive.org/web/20201230182007/https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf
+    # example 11.13
+    T = Matrix([[1, 0, 0, 0, 0],
+                [S(1) / 2, 0, S(1) / 2, 0, 0],
+                [0, S(1) / 2, 0, S(1) / 2, 0],
+                [0, 0, S(1) / 2, 0, S(1) / 2],
+                [0, 0, 0, 0, S(1)]])
+    DW = DiscreteMarkovChain('DW', [0, 1, 2, 3, 4], T)
+    states, A, B, C = DW.decompose()
+    assert states == [0, 4, 1, 2, 3]
+    assert A == Matrix([[1, 0], [0, 1]])
+    assert B == Matrix([[S(1)/2, 0], [0, 0], [0, S(1)/2]])
+    assert C == Matrix([[0, S(1)/2, 0], [S(1)/2, 0, S(1)/2], [0, S(1)/2, 0]])
+    states, new_matrix = DW.canonical_form()
+    assert states == [0, 4, 1, 2, 3]
+    assert new_matrix == Matrix([[1, 0, 0, 0, 0],
+                                 [0, 1, 0, 0, 0],
+                                 [S(1)/2, 0, 0, S(1)/2, 0],
+                                 [0, 0, S(1)/2, 0, S(1)/2],
+                                 [0, S(1)/2, 0, S(1)/2, 0]])
+
+    # test regular and ergodic
+    # https://web.archive.org/web/20201230182007/https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf
+    T = Matrix([[0, 4, 0, 0, 0],
+                [1, 0, 3, 0, 0],
+                [0, 2, 0, 2, 0],
+                [0, 0, 3, 0, 1],
+                [0, 0, 0, 4, 0]])/4
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert not X.is_regular()
+    assert X.is_ergodic()
+    T = Matrix([[0, 1], [1, 0]])
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert not X.is_regular()
+    assert X.is_ergodic()
+    # http://www.math.wisc.edu/~valko/courses/331/MC2.pdf
+    T = Matrix([[2, 1, 1],
+                [2, 0, 2],
+                [1, 1, 2]])/4
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert X.is_regular()
+    assert X.is_ergodic()
+    # https://docs.ufpr.br/~lucambio/CE222/1S2014/Kemeny-Snell1976.pdf
+    T = Matrix([[1, 1], [1, 1]])/2
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert X.is_regular()
+    assert X.is_ergodic()
+
+    # test is_absorbing_chain
+    T = Matrix([[0, 1, 0],
+                [1, 0, 0],
+                [0, 0, 1]])
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert not X.is_absorbing_chain()
+    # https://en.wikipedia.org/wiki/Absorbing_Markov_chain
+    T = Matrix([[1, 1, 0, 0],
+                [0, 1, 1, 0],
+                [1, 0, 0, 1],
+                [0, 0, 0, 2]])/2
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert X.is_absorbing_chain()
+    T = Matrix([[2, 0, 0, 0, 0],
+                [1, 0, 1, 0, 0],
+                [0, 1, 0, 1, 0],
+                [0, 0, 1, 0, 1],
+                [0, 0, 0, 0, 2]])/2
+    X = DiscreteMarkovChain('X', trans_probs=T)
+    assert X.is_absorbing_chain()
+
+    # test custom state space
+    Y10 = DiscreteMarkovChain('Y', [1, 2, 3], TO2)
+    tuples = Y10.communication_classes()
+    classes, recurrence, periods = list(zip(*tuples))
+    assert classes == ([1], [2, 3])
+    assert recurrence == (True, False)
+    assert periods == (1, 1)
+    assert Y10.canonical_form() == ([1, 2, 3], TO2)
+    assert Y10.decompose() == ([1, 2, 3], TO2[0:1, 0:1], TO2[1:3, 0:1], TO2[1:3, 1:3])
+
+    # testing miscellaneous queries
+    T = Matrix([[S.Half, Rational(1, 4), Rational(1, 4)],
+                [Rational(1, 3), 0, Rational(2, 3)],
+                [S.Half, S.Half, 0]])
+    X = DiscreteMarkovChain('X', [0, 1, 2], T)
+    assert P(Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
+    Eq(P(Eq(X[1], 0)), Rational(1, 4)) & Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
+    assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
+    assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
+    assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
+    assert E(X[1]**2, Eq(X[0], 1)) == Rational(8, 3)
+    assert variance(X[1], Eq(X[0], 1)) == Rational(8, 9)
+    raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
+    raises(ValueError, lambda: DiscreteMarkovChain('X', [0, 1], T))
+
+    # testing miscellaneous queries with different state space
+    X = DiscreteMarkovChain('X', ['A', 'B', 'C'], T)
+    assert P(Eq(X[1], 2) & Eq(X[2], 1) & Eq(X[3], 0),
+    Eq(P(Eq(X[1], 0)), Rational(1, 4)) & Eq(P(Eq(X[1], 1)), Rational(1, 4))) == Rational(1, 12)
+    assert P(Eq(X[2], 1) | Eq(X[2], 2), Eq(X[1], 1)) == Rational(2, 3)
+    assert P(Eq(X[2], 1) & Eq(X[2], 2), Eq(X[1], 1)) is S.Zero
+    assert P(Ne(X[2], 2), Eq(X[1], 1)) == Rational(1, 3)
+    a = X.state_space.args[0]
+    c = X.state_space.args[2]
+    assert (E(X[1] ** 2, Eq(X[0], 1)) - (a**2/3 + 2*c**2/3)).simplify() == 0
+    assert (variance(X[1], Eq(X[0], 1)) - (2*(-a/3 + c/3)**2/3 + (2*a/3 - 2*c/3)**2/3)).simplify() == 0
+    raises(ValueError, lambda: E(X[1], Eq(X[2], 1)))
+
+    #testing queries with multiple RandomIndexedSymbols
+    T = Matrix([[Rational(5, 10), Rational(3, 10), Rational(2, 10)], [Rational(2, 10), Rational(7, 10), Rational(1, 10)], [Rational(3, 10), Rational(3, 10), Rational(4, 10)]])
+    Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
+    assert P(Eq(Y[7], Y[5]), Eq(Y[2], 0)).round(5) == Float(0.44428, 5)
+    assert P(Gt(Y[3], Y[1]), Eq(Y[0], 0)).round(2) == Float(0.36, 2)
+    assert P(Le(Y[5], Y[10]), Eq(Y[4], 2)).round(6) == Float(0.583120, 6)
+    assert Float(P(Eq(Y[10], Y[5]), Eq(Y[4], 1)), 14) == Float(1 - P(Ne(Y[10], Y[5]), Eq(Y[4], 1)), 14)
+    assert Float(P(Gt(Y[8], Y[9]), Eq(Y[3], 2)), 14) == Float(1 - P(Le(Y[8], Y[9]), Eq(Y[3], 2)), 14)
+    assert Float(P(Lt(Y[1], Y[4]), Eq(Y[0], 0)), 14) == Float(1 - P(Ge(Y[1], Y[4]), Eq(Y[0], 0)), 14)
+    assert P(Eq(Y[5], Y[10]), Eq(Y[2], 1)) == P(Eq(Y[10], Y[5]), Eq(Y[2], 1))
+    assert P(Gt(Y[1], Y[2]), Eq(Y[0], 1)) == P(Lt(Y[2], Y[1]), Eq(Y[0], 1))
+    assert P(Ge(Y[7], Y[6]), Eq(Y[4], 1)) == P(Le(Y[6], Y[7]), Eq(Y[4], 1))
+
+    #test symbolic queries
+    a, b, c, d = symbols('a b c d')
+    T = Matrix([[Rational(1, 10), Rational(4, 10), Rational(5, 10)], [Rational(3, 10), Rational(4, 10), Rational(3, 10)], [Rational(7, 10), Rational(2, 10), Rational(1, 10)]])
+    Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
+    query = P(Eq(Y[a], b), Eq(Y[c], d))
+    assert query.subs({a:10, b:2, c:5, d:1}).evalf().round(4) == P(Eq(Y[10], 2), Eq(Y[5], 1)).round(4)
+    assert query.subs({a:15, b:0, c:10, d:1}).evalf().round(4) == P(Eq(Y[15], 0), Eq(Y[10], 1)).round(4)
+    query_gt = P(Gt(Y[a], b), Eq(Y[c], d))
+    query_le = P(Le(Y[a], b), Eq(Y[c], d))
+    assert query_gt.subs({a:5, b:2, c:1, d:0}).evalf() + query_le.subs({a:5, b:2, c:1, d:0}).evalf() == 1.0
+    query_ge = P(Ge(Y[a], b), Eq(Y[c], d))
+    query_lt = P(Lt(Y[a], b), Eq(Y[c], d))
+    assert query_ge.subs({a:4, b:1, c:0, d:2}).evalf() + query_lt.subs({a:4, b:1, c:0, d:2}).evalf() == 1.0
+
+    #test issue 20078
+    assert (2*Y[1] + 3*Y[1]).simplify() == 5*Y[1]
+    assert (2*Y[1] - 3*Y[1]).simplify() == -Y[1]
+    assert (2*(0.25*Y[1])).simplify() == 0.5*Y[1]
+    assert ((2*Y[1]) * (0.25*Y[1])).simplify() == 0.5*Y[1]**2
+    assert (Y[1]**2 + Y[1]**3).simplify() == (Y[1] + 1)*Y[1]**2
+
+def test_sample_stochastic_process():
+    if not import_module('scipy'):
+        skip('SciPy Not installed. Skip sampling tests')
+    import random
+    random.seed(0)
+    numpy = import_module('numpy')
+    if numpy:
+        numpy.random.seed(0) # scipy uses numpy to sample so to set its seed
+    T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
+    Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
+    for samps in range(10):
+        assert next(sample_stochastic_process(Y)) in Y.state_space
+    Z = DiscreteMarkovChain("Z", ['1', 1, 0], T)
+    for samps in range(10):
+        assert next(sample_stochastic_process(Z)) in Z.state_space
+
+    T = Matrix([[S.Half, Rational(1, 4), Rational(1, 4)],
+                [Rational(1, 3), 0, Rational(2, 3)],
+                [S.Half, S.Half, 0]])
+    X = DiscreteMarkovChain('X', [0, 1, 2], T)
+    for samps in range(10):
+        assert next(sample_stochastic_process(X)) in X.state_space
+    W = DiscreteMarkovChain('W', [1, pi, oo], T)
+    for samps in range(10):
+        assert next(sample_stochastic_process(W)) in W.state_space
+
+
+def test_ContinuousMarkovChain():
+    T1 = Matrix([[S(-2), S(2), S.Zero],
+                 [S.Zero, S.NegativeOne, S.One],
+                 [Rational(3, 2), Rational(3, 2), S(-3)]])
+    C1 = ContinuousMarkovChain('C', [0, 1, 2], T1)
+    assert C1.limiting_distribution() == ImmutableMatrix([[Rational(3, 19), Rational(12, 19), Rational(4, 19)]])
+
+    T2 = Matrix([[-S.One, S.One, S.Zero], [S.One, -S.One, S.Zero], [S.Zero, S.One, -S.One]])
+    C2 = ContinuousMarkovChain('C', [0, 1, 2], T2)
+    A, t = C2.generator_matrix, symbols('t', positive=True)
+    assert C2.transition_probabilities(A)(t) == Matrix([[S.Half + exp(-2*t)/2, S.Half - exp(-2*t)/2, 0],
+                                                       [S.Half - exp(-2*t)/2, S.Half + exp(-2*t)/2, 0],
+                                                       [S.Half - exp(-t) + exp(-2*t)/2, S.Half - exp(-2*t)/2, exp(-t)]])
+    with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
+        assert P(Eq(C2(1), 1), Eq(C2(0), 1), evaluate=False) == Probability(Eq(C2(1), 1), Eq(C2(0), 1))
+    assert P(Eq(C2(1), 1), Eq(C2(0), 1)) == exp(-2)/2 + S.Half
+    assert P(Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 1),
+                Eq(P(Eq(C2(1), 0)), S.Half)) == (Rational(1, 4) - exp(-2)/4)*(exp(-2)/2 + S.Half)
+    assert P(Not(Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 2)) |
+                (Eq(C2(1), 0) & Eq(C2(2), 1) & Eq(C2(3), 2)),
+                Eq(P(Eq(C2(1), 0)), Rational(1, 4)) & Eq(P(Eq(C2(1), 1)), Rational(1, 4))) is S.One
+    assert E(C2(Rational(3, 2)), Eq(C2(0), 2)) == -exp(-3)/2 + 2*exp(Rational(-3, 2)) + S.Half
+    assert variance(C2(Rational(3, 2)), Eq(C2(0), 1)) == ((S.Half - exp(-3)/2)**2*(exp(-3)/2 + S.Half)
+                                                    + (Rational(-1, 2) - exp(-3)/2)**2*(S.Half - exp(-3)/2))
+    raises(KeyError, lambda: P(Eq(C2(1), 0), Eq(P(Eq(C2(1), 1)), S.Half)))
+    assert P(Eq(C2(1), 0), Eq(P(Eq(C2(5), 1)), S.Half)) == Probability(Eq(C2(1), 0))
+    TS1 = MatrixSymbol('G', 3, 3)
+    CS1 = ContinuousMarkovChain('C', [0, 1, 2], TS1)
+    A = CS1.generator_matrix
+    assert CS1.transition_probabilities(A)(t) == exp(t*A)
+
+    C3 = ContinuousMarkovChain('C', [Symbol('0'), Symbol('1'), Symbol('2')], T2)
+    assert P(Eq(C3(1), 1), Eq(C3(0), 1)) == exp(-2)/2 + S.Half
+    assert P(Eq(C3(1), Symbol('1')), Eq(C3(0), Symbol('1'))) == exp(-2)/2 + S.Half
+
+    #test probability queries
+    G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]])
+    C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G)
+    assert P(Eq(C(7.385), C(3.19)), Eq(C(0.862), 0)).round(5) == Float(0.35469, 5)
+    assert P(Gt(C(98.715), C(19.807)), Eq(C(11.314), 2)).round(5) == Float(0.32452, 5)
+    assert P(Le(C(5.9), C(10.112)), Eq(C(4), 1)).round(6) == Float(0.675214, 6)
+    assert Float(P(Eq(C(7.32), C(2.91)), Eq(C(2.63), 1)), 14) == Float(1 - P(Ne(C(7.32), C(2.91)), Eq(C(2.63), 1)), 14)
+    assert Float(P(Gt(C(3.36), C(1.101)), Eq(C(0.8), 2)), 14) == Float(1 - P(Le(C(3.36), C(1.101)), Eq(C(0.8), 2)), 14)
+    assert Float(P(Lt(C(4.9), C(2.79)), Eq(C(1.61), 0)), 14) == Float(1 - P(Ge(C(4.9), C(2.79)), Eq(C(1.61), 0)), 14)
+    assert P(Eq(C(5.243), C(10.912)), Eq(C(2.174), 1)) == P(Eq(C(10.912), C(5.243)), Eq(C(2.174), 1))
+    assert P(Gt(C(2.344), C(9.9)), Eq(C(1.102), 1)) == P(Lt(C(9.9), C(2.344)), Eq(C(1.102), 1))
+    assert P(Ge(C(7.87), C(1.008)), Eq(C(0.153), 1)) == P(Le(C(1.008), C(7.87)), Eq(C(0.153), 1))
+
+    #test symbolic queries
+    a, b, c, d = symbols('a b c d')
+    query = P(Eq(C(a), b), Eq(C(c), d))
+    assert query.subs({a:3.65, b:2, c:1.78, d:1}).evalf().round(10) == P(Eq(C(3.65), 2), Eq(C(1.78), 1)).round(10)
+    query_gt = P(Gt(C(a), b), Eq(C(c), d))
+    query_le = P(Le(C(a), b), Eq(C(c), d))
+    assert query_gt.subs({a:13.2, b:0, c:3.29, d:2}).evalf() + query_le.subs({a:13.2, b:0, c:3.29, d:2}).evalf() == 1.0
+    query_ge = P(Ge(C(a), b), Eq(C(c), d))
+    query_lt = P(Lt(C(a), b), Eq(C(c), d))
+    assert query_ge.subs({a:7.43, b:1, c:1.45, d:0}).evalf() + query_lt.subs({a:7.43, b:1, c:1.45, d:0}).evalf() == 1.0
+
+    #test issue 20078
+    assert (2*C(1) + 3*C(1)).simplify() == 5*C(1)
+    assert (2*C(1) - 3*C(1)).simplify() == -C(1)
+    assert (2*(0.25*C(1))).simplify() == 0.5*C(1)
+    assert (2*C(1) * 0.25*C(1)).simplify() == 0.5*C(1)**2
+    assert (C(1)**2 + C(1)**3).simplify() == (C(1) + 1)*C(1)**2
+
+def test_BernoulliProcess():
+
+    B = BernoulliProcess("B", p=0.6, success=1, failure=0)
+    assert B.state_space == FiniteSet(0, 1)
+    assert B.index_set == S.Naturals0
+    assert B.success == 1
+    assert B.failure == 0
+
+    X = BernoulliProcess("X", p=Rational(1,3), success='H', failure='T')
+    assert X.state_space == FiniteSet('H', 'T')
+    H, T = symbols("H,T")
+    assert E(X[1]+X[2]*X[3]) == H**2/9 + 4*H*T/9 + H/3 + 4*T**2/9 + 2*T/3
+
+    t, x = symbols('t, x', positive=True, integer=True)
+    assert isinstance(B[t], RandomIndexedSymbol)
+
+    raises(ValueError, lambda: BernoulliProcess("X", p=1.1, success=1, failure=0))
+    raises(NotImplementedError, lambda: B(t))
+
+    raises(IndexError, lambda: B[-3])
+    assert B.joint_distribution(B[3], B[9]) == JointDistributionHandmade(Lambda((B[3], B[9]),
+                Piecewise((0.6, Eq(B[3], 1)), (0.4, Eq(B[3], 0)), (0, True))
+                *Piecewise((0.6, Eq(B[9], 1)), (0.4, Eq(B[9], 0)), (0, True))))
+
+    assert B.joint_distribution(2, B[4]) == JointDistributionHandmade(Lambda((B[2], B[4]),
+                Piecewise((0.6, Eq(B[2], 1)), (0.4, Eq(B[2], 0)), (0, True))
+                *Piecewise((0.6, Eq(B[4], 1)), (0.4, Eq(B[4], 0)), (0, True))))
+
+    # Test for the sum distribution of Bernoulli Process RVs
+    Y = B[1] + B[2] + B[3]
+    assert P(Eq(Y, 0)).round(2) == Float(0.06, 1)
+    assert P(Eq(Y, 2)).round(2) == Float(0.43, 2)
+    assert P(Eq(Y, 4)).round(2) == 0
+    assert P(Gt(Y, 1)).round(2) == Float(0.65, 2)
+    # Test for independency of each Random Indexed variable
+    assert P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0) & Eq(B[4], 1)).round(2) == Float(0.06, 1)
+
+    assert E(2 * B[1] + B[2]).round(2) == Float(1.80, 3)
+    assert E(2 * B[1] + B[2] + 5).round(2) == Float(6.80, 3)
+    assert E(B[2] * B[4] + B[10]).round(2) == Float(0.96, 2)
+    assert E(B[2] > 0, Eq(B[1],1) & Eq(B[2],1)).round(2) == Float(0.60,2)
+    assert E(B[1]) == 0.6
+    assert P(B[1] > 0).round(2) == Float(0.60, 2)
+    assert P(B[1] < 1).round(2) == Float(0.40, 2)
+    assert P(B[1] > 0, B[2] <= 1).round(2) == Float(0.60, 2)
+    assert P(B[12] * B[5] > 0).round(2) == Float(0.36, 2)
+    assert P(B[12] * B[5] > 0, B[4] < 1).round(2) == Float(0.36, 2)
+    assert P(Eq(B[2], 1), B[2] > 0) == 1.0
+    assert P(Eq(B[5], 3)) == 0
+    assert P(Eq(B[1], 1), B[1] < 0) == 0
+    assert P(B[2] > 0, Eq(B[2], 1)) == 1
+    assert P(B[2] < 0, Eq(B[2], 1)) == 0
+    assert P(B[2] > 0, B[2]==7) == 0
+    assert P(B[5] > 0, B[5]) == BernoulliDistribution(0.6, 0, 1)
+    raises(ValueError, lambda: P(3))
+    raises(ValueError, lambda: P(B[3] > 0, 3))
+
+    # test issue 19456
+    expr = Sum(B[t], (t, 0, 4))
+    expr2 = Sum(B[t], (t, 1, 3))
+    expr3 = Sum(B[t]**2, (t, 1, 3))
+    assert expr.doit() == B[0] + B[1] + B[2] + B[3] + B[4]
+    assert expr2.doit() == Y
+    assert expr3.doit() == B[1]**2 + B[2]**2 + B[3]**2
+    assert B[2*t].free_symbols == {B[2*t], t}
+    assert B[4].free_symbols == {B[4]}
+    assert B[x*t].free_symbols == {B[x*t], x, t}
+
+    #test issue 20078
+    assert (2*B[t] + 3*B[t]).simplify() == 5*B[t]
+    assert (2*B[t] - 3*B[t]).simplify() == -B[t]
+    assert (2*(0.25*B[t])).simplify() == 0.5*B[t]
+    assert (2*B[t] * 0.25*B[t]).simplify() == 0.5*B[t]**2
+    assert (B[t]**2 + B[t]**3).simplify() == (B[t] + 1)*B[t]**2
+
+def test_PoissonProcess():
+    X = PoissonProcess("X", 3)
+    assert X.state_space == S.Naturals0
+    assert X.index_set == Interval(0, oo)
+    assert X.lamda == 3
+
+    t, d, x, y = symbols('t d x y', positive=True)
+    assert isinstance(X(t), RandomIndexedSymbol)
+    assert X.distribution(t) == PoissonDistribution(3*t)
+    with warns_deprecated_sympy():
+        X.distribution(X(t))
+    raises(ValueError, lambda: PoissonProcess("X", -1))
+    raises(NotImplementedError, lambda: X[t])
+    raises(IndexError, lambda: X(-5))
+
+    assert X.joint_distribution(X(2), X(3)) == JointDistributionHandmade(Lambda((X(2), X(3)),
+                6**X(2)*9**X(3)*exp(-15)/(factorial(X(2))*factorial(X(3)))))
+
+    assert X.joint_distribution(4, 6) == JointDistributionHandmade(Lambda((X(4), X(6)),
+                12**X(4)*18**X(6)*exp(-30)/(factorial(X(4))*factorial(X(6)))))
+
+    assert P(X(t) < 1) == exp(-3*t)
+    assert P(Eq(X(t), 0), Contains(t, Interval.Lopen(3, 5))) == exp(-6) # exp(-2*lamda)
+    res = P(Eq(X(t), 1), Contains(t, Interval.Lopen(3, 4)))
+    assert res == 3*exp(-3)
+
+    # Equivalent to P(Eq(X(t), 1))**4 because of non-overlapping intervals
+    assert P(Eq(X(t), 1) & Eq(X(d), 1) & Eq(X(x), 1) & Eq(X(y), 1), Contains(t, Interval.Lopen(0, 1))
+    & Contains(d, Interval.Lopen(1, 2)) & Contains(x, Interval.Lopen(2, 3))
+    & Contains(y, Interval.Lopen(3, 4))) == res**4
+
+    # Return Probability because of overlapping intervals
+    assert P(Eq(X(t), 2) & Eq(X(d), 3), Contains(t, Interval.Lopen(0, 2))
+    & Contains(d, Interval.Ropen(2, 4))) == \
+                Probability(Eq(X(d), 3) & Eq(X(t), 2), Contains(t, Interval.Lopen(0, 2))
+                & Contains(d, Interval.Ropen(2, 4)))
+
+    raises(ValueError, lambda: P(Eq(X(t), 2) & Eq(X(d), 3),
+    Contains(t, Interval.Lopen(0, 4)) & Contains(d, Interval.Lopen(3, oo)))) # no bound on d
+    assert P(Eq(X(3), 2)) == 81*exp(-9)/2
+    assert P(Eq(X(t), 2), Contains(t, Interval.Lopen(0, 5))) == 225*exp(-15)/2
+
+    # Check that probability works correctly by adding it to 1
+    res1 = P(X(t) <= 3, Contains(t, Interval.Lopen(0, 5)))
+    res2 = P(X(t) > 3, Contains(t, Interval.Lopen(0, 5)))
+    assert res1 == 691*exp(-15)
+    assert (res1 + res2).simplify() == 1
+
+    # Check Not and  Or
+    assert P(Not(Eq(X(t), 2) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) & \
+            Contains(d, Interval.Lopen(7, 8))).simplify() == -18*exp(-6) + 234*exp(-9) + 1
+    assert P(Eq(X(t), 2) | Ne(X(t), 4), Contains(t, Interval.Ropen(2, 4))) == 1 - 36*exp(-6)
+    raises(ValueError, lambda: P(X(t) > 2, X(t) + X(d)))
+    assert E(X(t)) == 3*t  # property of the distribution at a given timestamp
+    assert E(X(t)**2 + X(d)*2 + X(y)**3, Contains(t, Interval.Lopen(0, 1))
+        & Contains(d, Interval.Lopen(1, 2)) & Contains(y, Interval.Ropen(3, 4))) == 75
+    assert E(X(t)**2, Contains(t, Interval.Lopen(0, 1))) == 12
+    assert E(x*(X(t) + X(d))*(X(t)**2+X(d)**2), Contains(t, Interval.Lopen(0, 1))
+    & Contains(d, Interval.Ropen(1, 2))) == \
+            Expectation(x*(X(d) + X(t))*(X(d)**2 + X(t)**2), Contains(t, Interval.Lopen(0, 1))
+            & Contains(d, Interval.Ropen(1, 2)))
+
+    # Value Error because of infinite time bound
+    raises(ValueError, lambda: E(X(t)**3, Contains(t, Interval.Lopen(1, oo))))
+
+    # Equivalent to E(X(t)**2) - E(X(d)**2) == E(X(1)**2) - E(X(1)**2) == 0
+    assert E((X(t) + X(d))*(X(t) - X(d)), Contains(t, Interval.Lopen(0, 1))
+        & Contains(d, Interval.Lopen(1, 2))) == 0
+    assert E(X(2) + x*E(X(5))) == 15*x + 6
+    assert E(x*X(1) + y) == 3*x + y
+    assert P(Eq(X(1), 2) & Eq(X(t), 3), Contains(t, Interval.Lopen(1, 2))) == 81*exp(-6)/4
+    Y = PoissonProcess("Y", 6)
+    Z = X + Y
+    assert Z.lamda == X.lamda + Y.lamda == 9
+    raises(ValueError, lambda: X + 5) # should be added be only PoissonProcess instance
+    N, M = Z.split(4, 5)
+    assert N.lamda == 4
+    assert M.lamda == 5
+    raises(ValueError, lambda: Z.split(3, 2)) # 2+3 != 9
+
+    raises(ValueError, lambda :P(Eq(X(t), 0), Contains(t, Interval.Lopen(1, 3)) & Eq(X(1), 0)))
+    # check if it handles queries with two random variables in one args
+    res1 = P(Eq(N(3), N(5)))
+    assert res1 == P(Eq(N(t), 0), Contains(t, Interval(3, 5)))
+    res2 = P(N(3) > N(1))
+    assert res2 == P((N(t) > 0), Contains(t, Interval(1, 3)))
+    assert P(N(3) < N(1)) == 0 # condition is not possible
+    res3 = P(N(3) <= N(1)) # holds only for Eq(N(3), N(1))
+    assert res3 == P(Eq(N(t), 0), Contains(t, Interval(1, 3)))
+
+    # tests from https://www.probabilitycourse.com/chapter11/11_1_2_basic_concepts_of_the_poisson_process.php
+    X = PoissonProcess('X', 10) # 11.1
+    assert P(Eq(X(S(1)/3), 3) & Eq(X(1), 10)) == exp(-10)*Rational(8000000000, 11160261)
+    assert P(Eq(X(1), 1), Eq(X(S(1)/3), 3)) == 0
+    assert P(Eq(X(1), 10), Eq(X(S(1)/3), 3)) == P(Eq(X(S(2)/3), 7))
+
+    X = PoissonProcess('X', 2) # 11.2
+    assert P(X(S(1)/2) < 1) == exp(-1)
+    assert P(X(3) < 1, Eq(X(1), 0)) == exp(-4)
+    assert P(Eq(X(4), 3), Eq(X(2), 3)) == exp(-4)
+
+    X = PoissonProcess('X', 3)
+    assert P(Eq(X(2), 5) & Eq(X(1), 2)) == Rational(81, 4)*exp(-6)
+
+    # check few properties
+    assert P(X(2) <= 3, X(1)>=1) == 3*P(Eq(X(1), 0)) + 2*P(Eq(X(1), 1)) + P(Eq(X(1), 2))
+    assert P(X(2) <= 3, X(1) > 1) == 2*P(Eq(X(1), 0)) + 1*P(Eq(X(1), 1))
+    assert P(Eq(X(2), 5) & Eq(X(1), 2)) == P(Eq(X(1), 3))*P(Eq(X(1), 2))
+    assert P(Eq(X(3), 4), Eq(X(1), 3)) == P(Eq(X(2), 1))
+
+    #test issue 20078
+    assert (2*X(t) + 3*X(t)).simplify() == 5*X(t)
+    assert (2*X(t) - 3*X(t)).simplify() == -X(t)
+    assert (2*(0.25*X(t))).simplify() == 0.5*X(t)
+    assert (2*X(t) * 0.25*X(t)).simplify() == 0.5*X(t)**2
+    assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1)*X(t)**2
+
+def test_WienerProcess():
+    X = WienerProcess("X")
+    assert X.state_space == S.Reals
+    assert X.index_set == Interval(0, oo)
+
+    t, d, x, y = symbols('t d x y', positive=True)
+    assert isinstance(X(t), RandomIndexedSymbol)
+    assert X.distribution(t) == NormalDistribution(0, sqrt(t))
+    with warns_deprecated_sympy():
+        X.distribution(X(t))
+    raises(ValueError, lambda: PoissonProcess("X", -1))
+    raises(NotImplementedError, lambda: X[t])
+    raises(IndexError, lambda: X(-2))
+
+    assert X.joint_distribution(X(2), X(3)) == JointDistributionHandmade(
+        Lambda((X(2), X(3)), sqrt(6)*exp(-X(2)**2/4)*exp(-X(3)**2/6)/(12*pi)))
+    assert X.joint_distribution(4, 6) == JointDistributionHandmade(
+        Lambda((X(4), X(6)), sqrt(6)*exp(-X(4)**2/8)*exp(-X(6)**2/12)/(24*pi)))
+
+    assert P(X(t) < 3).simplify() == erf(3*sqrt(2)/(2*sqrt(t)))/2 + S(1)/2
+    assert P(X(t) > 2, Contains(t, Interval.Lopen(3, 7))).simplify() == S(1)/2 -\
+                erf(sqrt(2)/2)/2
+
+    # Equivalent to P(X(1)>1)**4
+    assert P((X(t) > 4) & (X(d) > 3) & (X(x) > 2) & (X(y) > 1),
+        Contains(t, Interval.Lopen(0, 1)) & Contains(d, Interval.Lopen(1, 2))
+        & Contains(x, Interval.Lopen(2, 3)) & Contains(y, Interval.Lopen(3, 4))).simplify() ==\
+        (1 - erf(sqrt(2)/2))*(1 - erf(sqrt(2)))*(1 - erf(3*sqrt(2)/2))*(1 - erf(2*sqrt(2)))/16
+
+    # Contains an overlapping interval so, return Probability
+    assert P((X(t)< 2) & (X(d)> 3), Contains(t, Interval.Lopen(0, 2))
+        & Contains(d, Interval.Ropen(2, 4))) == Probability((X(d) > 3) & (X(t) < 2),
+        Contains(d, Interval.Ropen(2, 4)) & Contains(t, Interval.Lopen(0, 2)))
+
+    assert str(P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) &
+        Contains(d, Interval.Lopen(7, 8))).simplify()) == \
+                '-(1 - erf(3*sqrt(2)/2))*(2 - erfc(5/2))/4 + 1'
+    # Distribution has mean 0 at each timestamp
+    assert E(X(t)) == 0
+    assert E(x*(X(t) + X(d))*(X(t)**2+X(d)**2), Contains(t, Interval.Lopen(0, 1))
+    & Contains(d, Interval.Ropen(1, 2))) == Expectation(x*(X(d) + X(t))*(X(d)**2 + X(t)**2),
+    Contains(d, Interval.Ropen(1, 2)) & Contains(t, Interval.Lopen(0, 1)))
+    assert E(X(t) + x*E(X(3))) == 0
+
+    #test issue 20078
+    assert (2*X(t) + 3*X(t)).simplify() == 5*X(t)
+    assert (2*X(t) - 3*X(t)).simplify() == -X(t)
+    assert (2*(0.25*X(t))).simplify() == 0.5*X(t)
+    assert (2*X(t) * 0.25*X(t)).simplify() == 0.5*X(t)**2
+    assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1)*X(t)**2
+
+
+def test_GammaProcess_symbolic():
+    t, d, x, y, g, l = symbols('t d x y g l', positive=True)
+    X = GammaProcess("X", l, g)
+
+    raises(NotImplementedError, lambda: X[t])
+    raises(IndexError, lambda: X(-1))
+    assert isinstance(X(t), RandomIndexedSymbol)
+    assert X.state_space == Interval(0, oo)
+    assert X.distribution(t) == GammaDistribution(g*t, 1/l)
+    with warns_deprecated_sympy():
+        X.distribution(X(t))
+    assert X.joint_distribution(5, X(3)) == JointDistributionHandmade(Lambda(
+        (X(5), X(3)), l**(8*g)*exp(-l*X(3))*exp(-l*X(5))*X(3)**(3*g - 1)*X(5)**(5*g
+        - 1)/(gamma(3*g)*gamma(5*g))))
+    # property of the gamma process at any given timestamp
+    assert E(X(t)) == g*t/l
+    assert variance(X(t)).simplify() == g*t/l**2
+
+    # Equivalent to E(2*X(1)) + E(X(1)**2) + E(X(1)**3), where E(X(1)) == g/l
+    assert E(X(t)**2 + X(d)*2 + X(y)**3, Contains(t, Interval.Lopen(0, 1))
+        & Contains(d, Interval.Lopen(1, 2)) & Contains(y, Interval.Ropen(3, 4))) == \
+            2*g/l + (g**2 + g)/l**2 + (g**3 + 3*g**2 + 2*g)/l**3
+
+    assert P(X(t) > 3, Contains(t, Interval.Lopen(3, 4))).simplify() == \
+                                1 - lowergamma(g, 3*l)/gamma(g) # equivalent to P(X(1)>3)
+
+
+    #test issue 20078
+    assert (2*X(t) + 3*X(t)).simplify() == 5*X(t)
+    assert (2*X(t) - 3*X(t)).simplify() == -X(t)
+    assert (2*(0.25*X(t))).simplify() == 0.5*X(t)
+    assert (2*X(t) * 0.25*X(t)).simplify() == 0.5*X(t)**2
+    assert (X(t)**2 + X(t)**3).simplify() == (X(t) + 1)*X(t)**2
+def test_GammaProcess_numeric():
+    t, d, x, y = symbols('t d x y', positive=True)
+    X = GammaProcess("X", 1, 2)
+    assert X.state_space == Interval(0, oo)
+    assert X.index_set == Interval(0, oo)
+    assert X.lamda == 1
+    assert X.gamma == 2
+
+    raises(ValueError, lambda: GammaProcess("X", -1, 2))
+    raises(ValueError, lambda: GammaProcess("X", 0, -2))
+    raises(ValueError, lambda: GammaProcess("X", -1, -2))
+
+    # all are independent because of non-overlapping intervals
+    assert P((X(t) > 4) & (X(d) > 3) & (X(x) > 2) & (X(y) > 1), Contains(t,
+        Interval.Lopen(0, 1)) & Contains(d, Interval.Lopen(1, 2)) & Contains(x,
+        Interval.Lopen(2, 3)) & Contains(y, Interval.Lopen(3, 4))).simplify() == \
+                                                            120*exp(-10)
+
+    # Check working with Not and Or
+    assert P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) &
+        Contains(d, Interval.Lopen(7, 8))).simplify() == -4*exp(-3) + 472*exp(-8)/3 + 1
+    assert P((X(t) > 2) | (X(t) < 4), Contains(t, Interval.Ropen(1, 4))).simplify() == \
+                                            -643*exp(-4)/15 + 109*exp(-2)/15 + 1
+
+    assert E(X(t)) == 2*t # E(X(t)) == gamma*t/l
+    assert E(X(2) + x*E(X(5))) == 10*x + 4

rl/Lib/site-packages/sympy/stats/tests/test_symbolic_multivariate.py

@@ -0,0 +1,172 @@
+from sympy.stats import Expectation, Normal, Variance, Covariance
+from sympy.testing.pytest import raises
+from sympy.core.symbol import symbols
+from sympy.matrices.exceptions import ShapeError
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.expressions.special import ZeroMatrix
+from sympy.stats.rv import RandomMatrixSymbol
+from sympy.stats.symbolic_multivariate_probability import (ExpectationMatrix,
+                            VarianceMatrix, CrossCovarianceMatrix)
+
+j, k = symbols("j,k")
+
+A = MatrixSymbol("A", k, k)
+B = MatrixSymbol("B", k, k)
+C = MatrixSymbol("C", k, k)
+D = MatrixSymbol("D", k, k)
+
+a = MatrixSymbol("a", k, 1)
+b = MatrixSymbol("b", k, 1)
+
+A2 = MatrixSymbol("A2", 2, 2)
+B2 = MatrixSymbol("B2", 2, 2)
+
+X = RandomMatrixSymbol("X", k, 1)
+Y = RandomMatrixSymbol("Y", k, 1)
+Z = RandomMatrixSymbol("Z", k, 1)
+W = RandomMatrixSymbol("W", k, 1)
+
+R = RandomMatrixSymbol("R", k, k)
+
+X2 = RandomMatrixSymbol("X2", 2, 1)
+
+normal = Normal("normal", 0, 1)
+
+m1 = Matrix([
+    [1, j*Normal("normal2", 2, 1)],
+    [normal, 0]
+])
+
+def test_multivariate_expectation():
+    expr = Expectation(a)
+    assert expr == Expectation(a) == ExpectationMatrix(a)
+    assert expr.expand() == a
+
+    expr = Expectation(X)
+    assert expr == Expectation(X) == ExpectationMatrix(X)
+    assert expr.shape == (k, 1)
+    assert expr.rows == k
+    assert expr.cols == 1
+    assert isinstance(expr, ExpectationMatrix)
+
+    expr = Expectation(A*X + b)
+    assert expr == ExpectationMatrix(A*X + b)
+    assert expr.expand() == A*ExpectationMatrix(X) + b
+    assert isinstance(expr, ExpectationMatrix)
+    assert expr.shape == (k, 1)
+
+    expr = Expectation(m1*X2)
+    assert expr.expand() == expr
+
+    expr = Expectation(A2*m1*B2*X2)
+    assert expr.args[0].args == (A2, m1, B2, X2)
+    assert expr.expand() == A2*ExpectationMatrix(m1*B2*X2)
+
+    expr = Expectation((X + Y)*(X - Y).T)
+    assert expr.expand() == ExpectationMatrix(X*X.T) - ExpectationMatrix(X*Y.T) +\
+                ExpectationMatrix(Y*X.T) - ExpectationMatrix(Y*Y.T)
+
+    expr = Expectation(A*X + B*Y)
+    assert expr.expand() == A*ExpectationMatrix(X) + B*ExpectationMatrix(Y)
+
+    assert Expectation(m1).doit() == Matrix([[1, 2*j], [0, 0]])
+
+    x1 = Matrix([
+    [Normal('N11', 11, 1), Normal('N12', 12, 1)],
+    [Normal('N21', 21, 1), Normal('N22', 22, 1)]
+    ])
+    x2 = Matrix([
+    [Normal('M11', 1, 1), Normal('M12', 2, 1)],
+    [Normal('M21', 3, 1), Normal('M22', 4, 1)]
+    ])
+
+    assert Expectation(Expectation(x1 + x2)).doit(deep=False) == ExpectationMatrix(x1 + x2)
+    assert Expectation(Expectation(x1 + x2)).doit() == Matrix([[12, 14], [24, 26]])
+
+
+def test_multivariate_variance():
+    raises(ShapeError, lambda: Variance(A))
+
+    expr = Variance(a)
+    assert expr == Variance(a) == VarianceMatrix(a)
+    assert expr.expand() == ZeroMatrix(k, k)
+    expr = Variance(a.T)
+    assert expr == Variance(a.T) == VarianceMatrix(a.T)
+    assert expr.expand() == ZeroMatrix(k, k)
+
+    expr = Variance(X)
+    assert expr == Variance(X) == VarianceMatrix(X)
+    assert expr.shape == (k, k)
+    assert expr.rows == k
+    assert expr.cols == k
+    assert isinstance(expr, VarianceMatrix)
+
+    expr = Variance(A*X)
+    assert expr == VarianceMatrix(A*X)
+    assert expr.expand() == A*VarianceMatrix(X)*A.T
+    assert isinstance(expr, VarianceMatrix)
+    assert expr.shape == (k, k)
+
+    expr = Variance(A*B*X)
+    assert expr.expand() == A*B*VarianceMatrix(X)*B.T*A.T
+
+    expr = Variance(m1*X2)
+    assert expr.expand() == expr
+
+    expr = Variance(A2*m1*B2*X2)
+    assert expr.args[0].args == (A2, m1, B2, X2)
+    assert expr.expand() == expr
+
+    expr = Variance(A*X + B*Y)
+    assert expr.expand() == 2*A*CrossCovarianceMatrix(X, Y)*B.T +\
+                    A*VarianceMatrix(X)*A.T + B*VarianceMatrix(Y)*B.T
+
+def test_multivariate_crosscovariance():
+    raises(ShapeError, lambda: Covariance(X, Y.T))
+    raises(ShapeError, lambda: Covariance(X, A))
+
+
+    expr = Covariance(a.T, b.T)
+    assert expr.shape == (1, 1)
+    assert expr.expand() == ZeroMatrix(1, 1)
+
+    expr = Covariance(a, b)
+    assert expr == Covariance(a, b) == CrossCovarianceMatrix(a, b)
+    assert expr.expand() == ZeroMatrix(k, k)
+    assert expr.shape == (k, k)
+    assert expr.rows == k
+    assert expr.cols == k
+    assert isinstance(expr, CrossCovarianceMatrix)
+
+    expr = Covariance(A*X + a, b)
+    assert expr.expand() == ZeroMatrix(k, k)
+
+    expr = Covariance(X, Y)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == expr
+
+    expr = Covariance(X, X)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == VarianceMatrix(X)
+
+    expr = Covariance(X + Y, Z)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == CrossCovarianceMatrix(X, Z) + CrossCovarianceMatrix(Y, Z)
+
+    expr = Covariance(A*X, Y)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == A*CrossCovarianceMatrix(X, Y)
+
+    expr = Covariance(X, B*Y)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == CrossCovarianceMatrix(X, Y)*B.T
+
+    expr = Covariance(A*X + a, B.T*Y + b)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == A*CrossCovarianceMatrix(X, Y)*B
+
+    expr = Covariance(A*X + B*Y + a, C.T*Z + D.T*W + b)
+    assert isinstance(expr, CrossCovarianceMatrix)
+    assert expr.expand() == A*CrossCovarianceMatrix(X, W)*D + A*CrossCovarianceMatrix(X, Z)*C \
+        + B*CrossCovarianceMatrix(Y, W)*D + B*CrossCovarianceMatrix(Y, Z)*C

rl/Lib/site-packages/sympy/stats/tests/test_symbolic_probability.py

@@ -0,0 +1,175 @@
+from sympy.concrete.summations import Sum
+from sympy.core.mul import Mul
+from sympy.core.numbers import (oo, pi)
+from sympy.core.relational import Eq
+from sympy.core.symbol import (Dummy, symbols)
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin
+from sympy.integrals.integrals import Integral
+from sympy.core.expr import unchanged
+from sympy.stats import (Normal, Poisson, variance, Covariance, Variance,
+                         Probability, Expectation, Moment, CentralMoment)
+from sympy.stats.rv import probability, expectation
+
+
+def test_literal_probability():
+    X = Normal('X', 2, 3)
+    Y = Normal('Y', 3, 4)
+    Z = Poisson('Z', 4)
+    W = Poisson('W', 3)
+    x = symbols('x', real=True)
+    y, w, z = symbols('y, w, z')
+
+    assert Probability(X > 0).evaluate_integral() == probability(X > 0)
+    assert Probability(X > x).evaluate_integral() == probability(X > x)
+    assert Probability(X > 0).rewrite(Integral).doit() == probability(X > 0)
+    assert Probability(X > x).rewrite(Integral).doit() == probability(X > x)
+
+    assert Expectation(X).evaluate_integral() == expectation(X)
+    assert Expectation(X).rewrite(Integral).doit() == expectation(X)
+    assert Expectation(X**2).evaluate_integral() == expectation(X**2)
+    assert Expectation(x*X).args == (x*X,)
+    assert Expectation(x*X).expand() == x*Expectation(X)
+    assert Expectation(2*X + 3*Y + z*X*Y).expand() == 2*Expectation(X) + 3*Expectation(Y) + z*Expectation(X*Y)
+    assert Expectation(2*X + 3*Y + z*X*Y).args == (2*X + 3*Y + z*X*Y,)
+    assert Expectation(sin(X)) == Expectation(sin(X)).expand()
+    assert Expectation(2*x*sin(X)*Y + y*X**2 + z*X*Y).expand() == 2*x*Expectation(sin(X)*Y) \
+                            + y*Expectation(X**2) + z*Expectation(X*Y)
+    assert Expectation(X + Y).expand() ==  Expectation(X) + Expectation(Y)
+    assert Expectation((X + Y)*(X - Y)).expand() == Expectation(X**2) - Expectation(Y**2)
+    assert Expectation((X + Y)*(X - Y)).expand().doit() == -12
+    assert Expectation(X + Y, evaluate=True).doit() == 5
+    assert Expectation(X + Expectation(Y)).doit() == 5
+    assert Expectation(X + Expectation(Y)).doit(deep=False) == 2 + Expectation(Expectation(Y))
+    assert Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False) == 2 \
+                                + Expectation(Expectation(Y + Expectation(2*X)))
+    assert Expectation(X + Expectation(Y + Expectation(2*X))).doit() == 9
+    assert Expectation(Expectation(2*X)).doit() == 4
+    assert Expectation(Expectation(2*X)).doit(deep=False) == Expectation(2*X)
+    assert Expectation(4*Expectation(2*X)).doit(deep=False) == 4*Expectation(2*X)
+    assert Expectation((X + Y)**3).expand() == 3*Expectation(X*Y**2) +\
+                3*Expectation(X**2*Y) + Expectation(X**3) + Expectation(Y**3)
+    assert Expectation((X - Y)**3).expand() == 3*Expectation(X*Y**2) -\
+                3*Expectation(X**2*Y) + Expectation(X**3) - Expectation(Y**3)
+    assert Expectation((X - Y)**2).expand() == -2*Expectation(X*Y) +\
+                Expectation(X**2) + Expectation(Y**2)
+
+    assert Variance(w).args == (w,)
+    assert Variance(w).expand() == 0
+    assert Variance(X).evaluate_integral() == Variance(X).rewrite(Integral).doit() == variance(X)
+    assert Variance(X + z).args == (X + z,)
+    assert Variance(X + z).expand() == Variance(X)
+    assert Variance(X*Y).args == (Mul(X, Y),)
+    assert type(Variance(X*Y)) == Variance
+    assert Variance(z*X).expand() == z**2*Variance(X)
+    assert Variance(X + Y).expand() == Variance(X) + Variance(Y) + 2*Covariance(X, Y)
+    assert Variance(X + Y + Z + W).expand() == (Variance(X) + Variance(Y) + Variance(Z) + Variance(W) +
+                                       2 * Covariance(X, Y) + 2 * Covariance(X, Z) + 2 * Covariance(X, W) +
+                                       2 * Covariance(Y, Z) + 2 * Covariance(Y, W) + 2 * Covariance(W, Z))
+    assert Variance(X**2).evaluate_integral() == variance(X**2)
+    assert unchanged(Variance, X**2)
+    assert Variance(x*X**2).expand() == x**2*Variance(X**2)
+    assert Variance(sin(X)).args == (sin(X),)
+    assert Variance(sin(X)).expand() == Variance(sin(X))
+    assert Variance(x*sin(X)).expand() == x**2*Variance(sin(X))
+
+    assert Covariance(w, z).args == (w, z)
+    assert Covariance(w, z).expand() == 0
+    assert Covariance(X, w).expand() == 0
+    assert Covariance(w, X).expand() == 0
+    assert Covariance(X, Y).args == (X, Y)
+    assert type(Covariance(X, Y)) == Covariance
+    assert Covariance(z*X + 3, Y).expand() == z*Covariance(X, Y)
+    assert Covariance(X, X).args == (X, X)
+    assert Covariance(X, X).expand() == Variance(X)
+    assert Covariance(z*X + 3, w*Y + 4).expand() == w*z*Covariance(X,Y)
+    assert Covariance(X, Y) == Covariance(Y, X)
+    assert Covariance(X + Y, Z + W).expand() == Covariance(W, X) + Covariance(W, Y) + Covariance(X, Z) + Covariance(Y, Z)
+    assert Covariance(x*X + y*Y, z*Z + w*W).expand() == (x*w*Covariance(W, X) + w*y*Covariance(W, Y) +
+                                                x*z*Covariance(X, Z) + y*z*Covariance(Y, Z))
+    assert Covariance(x*X**2 + y*sin(Y), z*Y*Z**2 + w*W).expand() == (w*x*Covariance(W, X**2) + w*y*Covariance(sin(Y), W) +
+                                                        x*z*Covariance(Y*Z**2, X**2) + y*z*Covariance(Y*Z**2, sin(Y)))
+    assert Covariance(X, X**2).expand() == Covariance(X, X**2)
+    assert Covariance(X, sin(X)).expand() == Covariance(sin(X), X)
+    assert Covariance(X**2, sin(X)*Y).expand() == Covariance(sin(X)*Y, X**2)
+    assert Covariance(w, X).evaluate_integral() == 0
+
+
+def test_probability_rewrite():
+    X = Normal('X', 2, 3)
+    Y = Normal('Y', 3, 4)
+    Z = Poisson('Z', 4)
+    W = Poisson('W', 3)
+    x, y, w, z = symbols('x, y, w, z')
+
+    assert Variance(w).rewrite(Expectation) == 0
+    assert Variance(X).rewrite(Expectation) == Expectation(X ** 2) - Expectation(X) ** 2
+    assert Variance(X, condition=Y).rewrite(Expectation) == Expectation(X ** 2, Y) - Expectation(X, Y) ** 2
+    assert Variance(X, Y) != Expectation(X**2) - Expectation(X)**2
+    assert Variance(X + z).rewrite(Expectation) == Expectation((X + z) ** 2) - Expectation(X + z) ** 2
+    assert Variance(X * Y).rewrite(Expectation) == Expectation(X ** 2 * Y ** 2) - Expectation(X * Y) ** 2
+
+    assert Covariance(w, X).rewrite(Expectation) == -w*Expectation(X) + Expectation(w*X)
+    assert Covariance(X, Y).rewrite(Expectation) == Expectation(X*Y) - Expectation(X)*Expectation(Y)
+    assert Covariance(X, Y, condition=W).rewrite(Expectation) == Expectation(X * Y, W) - Expectation(X, W) * Expectation(Y, W)
+
+    w, x, z = symbols("W, x, z")
+    px = Probability(Eq(X, x))
+    pz = Probability(Eq(Z, z))
+
+    assert Expectation(X).rewrite(Probability) == Integral(x*px, (x, -oo, oo))
+    assert Expectation(Z).rewrite(Probability) == Sum(z*pz, (z, 0, oo))
+    assert Variance(X).rewrite(Probability) == Integral(x**2*px, (x, -oo, oo)) - Integral(x*px, (x, -oo, oo))**2
+    assert Variance(Z).rewrite(Probability) == Sum(z**2*pz, (z, 0, oo)) - Sum(z*pz, (z, 0, oo))**2
+    assert Covariance(w, X).rewrite(Probability) == \
+           -w*Integral(x*Probability(Eq(X, x)), (x, -oo, oo)) + Integral(w*x*Probability(Eq(X, x)), (x, -oo, oo))
+
+    # To test rewrite as sum function
+    assert Variance(X).rewrite(Sum) == Variance(X).rewrite(Integral)
+    assert Expectation(X).rewrite(Sum) == Expectation(X).rewrite(Integral)
+
+    assert Covariance(w, X).rewrite(Sum) == 0
+
+    assert Covariance(w, X).rewrite(Integral) == 0
+
+    assert Variance(X, condition=Y).rewrite(Probability) == Integral(x**2*Probability(Eq(X, x), Y), (x, -oo, oo)) - \
+                                                            Integral(x*Probability(Eq(X, x), Y), (x, -oo, oo))**2
+
+
+def test_symbolic_Moment():
+    mu = symbols('mu', real=True)
+    sigma = symbols('sigma', positive=True)
+    x = symbols('x')
+    X = Normal('X', mu, sigma)
+    M = Moment(X, 4, 2)
+    assert M.rewrite(Expectation) == Expectation((X - 2)**4)
+    assert M.rewrite(Probability) == Integral((x - 2)**4*Probability(Eq(X, x)),
+                                    (x, -oo, oo))
+    k = Dummy('k')
+    expri = Integral(sqrt(2)*(k - 2)**4*exp(-(k - \
+                mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo))
+    assert M.rewrite(Integral).dummy_eq(expri)
+    assert M.doit() == (mu**4 - 8*mu**3 + 6*mu**2*sigma**2 + \
+                24*mu**2 - 24*mu*sigma**2 - 32*mu + 3*sigma**4 + 24*sigma**2 + 16)
+    M = Moment(2, 5)
+    assert M.doit() == 2**5
+
+
+def test_symbolic_CentralMoment():
+    mu = symbols('mu', real=True)
+    sigma = symbols('sigma', positive=True)
+    x = symbols('x')
+    X = Normal('X', mu, sigma)
+    CM = CentralMoment(X, 6)
+    assert CM.rewrite(Expectation) == Expectation((X - Expectation(X))**6)
+    assert CM.rewrite(Probability) == Integral((x - Integral(x*Probability(True),
+                    (x, -oo, oo)))**6*Probability(Eq(X, x)), (x, -oo, oo))
+    k = Dummy('k')
+    expri = Integral(sqrt(2)*(k - Integral(sqrt(2)*k*exp(-(k - \
+        mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo)))**6*exp(-(k - \
+        mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (k, -oo, oo))
+    assert CM.rewrite(Integral).dummy_eq(expri)
+    assert CM.doit().simplify() == 15*sigma**6
+    CM = Moment(5, 5)
+    assert CM.doit() == 5**5