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) pytest.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_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_die_args(): pytest.raises(ValueError, lambda: Die('X', -1)) # issue sympy/sympy#8105: negative sides. pytest.raises(ValueError, lambda: Die('X', 0)) pytest.raises(ValueError, lambda: Die('X', 1.5)) # issue sympy/sympy#8103: non integer sides. k = Symbol('k') sym_die = Die('X', k) pytest.raises(ValueError, lambda: density(sym_die).dict)
def test_dice(): # TODO: Make iid method! X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) a, b = symbols('a b') assert E(X) == 3 + Rational(1, 2) 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) == 0 assert covariance(X, X + Y) == variance(X) assert density(Eq(cos(X * pi), 1))[True] == Rational(1, 2) assert correlation(X, Y) == 0 assert correlation(X, Y) == correlation(Y, X) assert smoment(X + Y, 3) == skewness(X + Y) assert smoment(X, 0) == 1 assert P(X > 3) == Rational(1, 2) assert P(2 * X > 6) == Rational(1, 2) 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 P(X > 3, X > 3) == 1 assert P(X > Y, Eq(Y, 6)) == 0 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[22] == Rational(1, 108) and d[4100] == Rational( 1, 216) and 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) X = Die('X', 2) x = X.symbol assert X.pspace.compute_cdf(X) == {1: Rational(1, 2), 2: 1} assert X.pspace.sorted_cdf(X) == [(1, Rational(1, 2)), (2, 1)] assert X.pspace.compute_density(X)(1) == Rational(1, 2) assert X.pspace.compute_density(X)(0) == 0 assert X.pspace.compute_density(X)(8) == 0 assert X.pspace.density == x
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_discreteuniform(): # Symbolic a, b, c = symbols('a b c') X = DiscreteUniform('X', [a, b, c]) assert X.pspace.distribution.pdf(a) == Rational(1, 3) assert X.pspace.distribution.pdf(p) == 0 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)) == Rational(1, 3) Y = DiscreteUniform('Y', range(-5, 5)) # Numeric assert E(Y) == -Rational(1, 2) assert variance(Y) == Rational(33, 4) for x in range(-5, 5): assert P(Eq(Y, x)) == Rational(1, 10) assert P(Y <= x) == Rational(x + 6, 10) assert P(Y >= x) == Rational(5 - x, 10) assert dict(density(Die('D', 6)).items()) == \ dict(density(DiscreteUniform('U', range(1, 7))).items())
def test_dice(): # TODO: Make iid method! X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) a, b = symbols('a b') 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) == 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, 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 P(X > 3, X > 3) == S.One assert P(X > Y, Eq(Y, 6)) == S.Zero assert P(Eq(X + Y, 12)) == S.One / 36 assert P(Eq(X + Y, 12), Eq(X, 6)) == S.One / 6 assert density(X + Y) == density(Y + Z) != density(X + X) d = density(2 * X + Y**Z) assert d[Integer(22)] == S.One / 108 and d[Integer( 4100)] == S.One / 216 and Integer(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)
def test_Density(): X = Die('X', 6) d = Density(X) assert d.doit() == density(X) r = Rational(1, 6) assert density(2*X).dict == {2: r, 4: r, 6: r, 8: r, 10: r, 12: r} X = Normal('X', 0, 1) Y = Normal('Y', 0, 1) assert str(density(X - Y, Y)(z)) == 'sqrt(2)*E**(-(Y + z)**2/2)/(2*sqrt(pi))'
def test_RandomDomain(): X = Normal('x1', 0, 1) assert str(where(X > 0)) == "Domain: (0 < x1) & (x1 < oo)" D = Die('d1', 6) assert str(where(D > 4)) == "Domain: (Eq(d1, 5)) | (Eq(d1, 6))" A = Exponential('a', 1) B = Exponential('b', 1) assert str(pspace(Tuple(A, B)).domain) == "Domain: (0 <= a) & (0 <= b) & (a < oo) & (b < oo)"
def test_RandomDomain(): X = Normal('x1', 0, 1) assert str(where(X > 0)) == "Domain: And(0 < x1, x1 < oo)" D = Die('d1', 6) assert str(where(D > 4)) == "Domain: Or(Eq(d1, 5), Eq(d1, 6))" A = Exponential('a', 1) B = Exponential('b', 1) assert str(pspace(Tuple( A, B)).domain) == "Domain: And(0 <= a, 0 <= b, a < oo, b < oo)"
def test_cdf(): D = Die('D', 6) o = Integer(1) 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_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 pytest.raises(TypeError): XX = given(X, X + 3)
def test_ProductPSpace(): X = Normal('X', 0, 1) Y = Normal('Y', 0, 1) px = X.pspace py = Y.pspace assert pspace(X + Y) == ProductPSpace(px, py) assert pspace(X + Y) == ProductPSpace(py, px) X = Die('X', 2) Y = Die('Y', 2) assert (pspace(X + Y).density == Dict((frozenset({('X', 1), ('Y', 2)}), Rational(1, 4)), (frozenset({('X', 1), ('Y', 1)}), Rational(1, 4)), (frozenset({('X', 2), ('Y', 1)}), Rational(1, 4)), (frozenset({('X', 2), ('Y', 2)}), Rational(1, 4)))) d = pspace(X + Y).domain assert ((X.symbol, 1), (Y.symbol, 2)) in d assert ((X.symbol, 0), (Y.symbol, 2)) not in d Z = Die('Z', 2) d1 = pspace(X + Y).domain assert ProductDomain(d1, Z.pspace.domain) == pspace(X + Y + Z).domain
def test_Sample(): X = Die('X', 6) Y = Normal('Y', 0, 1) z = Symbol('z') assert sample(X) in [1, 2, 3, 4, 5, 6] assert sample(X + Y).is_Float P(X + Y > 0, Y < 0, numsamples=10).is_number assert E(X + Y, numsamples=10).is_number assert variance(X + Y, numsamples=10).is_number pytest.raises(ValueError, lambda: P(Y > z, numsamples=5)) assert P(sin(Y) <= 1, numsamples=10) == 1 assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10) == 1 # Make sure this doesn't raise an error E(Sum(1 / z**Y, (z, 1, oo)), Y > 2, numsamples=3) 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))
def test_Density(): X = Die('X', 6) d = Density(X) assert d.doit() == density(X)
def test_given(): X = Die('X', 6) assert density(X, X > 5) == {6: 1} assert where(X > 2, X > 5).as_boolean() == Eq(X.symbol, 6) assert sample(X, X > 5) == 6
def test_dice_bayes(): X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) BayesTest(X > 3, X + Y < 5) BayesTest(Eq(X - Y, Z), Z > Y) BayesTest(X > 3, X > 2)