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)