def test_dice():
# TODO: Make iid method!
X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
a, b, t = symbols('a b t')
assert E(X) == 3 + S.Half
assert variance(X) == S(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) == S(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[S(22)] == S.One / 108 and d[S(4100)] == S.One / 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
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_sample_numpy():
distribs_numpy = [Binomial("B", 5, 0.4), Hypergeometric("H", 2, 1, 1)]
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 sam in X.pspace.domain.set
raises(NotImplementedError, lambda: sample(Die("D"), library='numpy'))
raises(NotImplementedError,
lambda: Die("D").pspace.sample(library='tensorflow'))
def test_sample_scipy():
distribs_scipy = [
FiniteRV('F', {
1: S.Half,
2: Rational(1, 4),
3: Rational(1, 4)
}),
DiscreteUniform("Y", list(range(5))),
Die("D"),
Bernoulli("Be", 0.3),
Binomial("Bi", 5, 0.4),
BetaBinomial("Bb", 2, 1, 1),
Hypergeometric("H", 1, 1, 1),
Rademacher("R")
]
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 sam in X.pspace.domain.set
for i in range(2):
for j in range(2):
assert samps2[i][j] in X.pspace.domain.set
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(1) / 2 + 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')
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert next(sample(X)) in [1, 2, 3, 4, 5, 6]
assert isinstance(next(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
assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10) == 1
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_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")
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)
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')
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
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_scipy():
distribs_scipy = [
FiniteRV('F', {1: S.Half, 2: Rational(1, 4), 3: Rational(1, 4)}),
DiscreteUniform("Y", list(range(5))),
Die("D"),
Bernoulli("Be", 0.3),
Binomial("Bi", 5, 0.4),
BetaBinomial("Bb", 2, 1, 1),
Hypergeometric("H", 1, 1, 1),
Rademacher("R")
]
size = 3
numsamples = 5
scipy = import_module('scipy')
if not scipy:
skip('Scipy not installed. Abort tests for _sample_scipy.')
else:
with ignore_warnings(UserWarning): ### TODO: Restore tests once warnings are removed
h_sample = list(sample(Hypergeometric("H", 1, 1, 1), size=size, numsamples=numsamples))
assert len(h_sample) == numsamples
for X in distribs_scipy:
samps = next(sample(X, size=size))
samps2 = next(sample(X, size=(2, 2)))
for sam in samps:
assert sam in X.pspace.domain.set
for i in range(2):
for j in range(2):
assert samps2[i][j] in X.pspace.domain.set
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_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
assert P(sin(Y) <= 1, cos(Y) < 1, numsamples=10) == 1
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)
def test_given_sample():
X = Die('X', 6)
scipy = import_module('scipy')
if not scipy:
skip('Scipy is not installed. Abort tests')
with ignore_warnings(
UserWarning): ### TODO: Restore tests once warnings are removed
assert next(sample(X, X > 5)) == 6
def test_sample_numpy():
distribs_numpy = [
Binomial("B", 5, 0.4),
]
size = 3
numpy = import_module('numpy')
if not numpy:
skip('Numpy is not installed. Abort tests for _sample_numpy.')
else:
with ignore_warnings(UserWarning):
for X in distribs_numpy:
samps = next(sample(X, size=size, library='numpy'))
for sam in samps:
assert sam in X.pspace.domain.set
raises(NotImplementedError,
lambda: next(sample(Die("D"), library='numpy')))
raises(NotImplementedError,
lambda: Die("D").pspace.sample(library='tensorflow'))
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)
scipy = import_module('scipy')
if not scipy:
skip('Scipy is not installed. Abort tests')
with ignore_warnings(UserWarning):
assert next(sample(X, X > 5)) == 6
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
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_where():
X, Y = Die(), Die()
Z = Normal(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(0, 1), Normal(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)
with raises(TypeError):
XX = given(X, X+3)
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert str(where(X > 0)) == "Domain: x1 > 0"
D = Die('d1', 6)
assert str(where(D > 4)) == "Domain: Or(d1 == 5, d1 == 6)"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert str(pspace(Tuple(A, B)).domain) == "Domain: And(a >= 0, b >= 0)"
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
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():
from sympy.stats import Normal, Die, Exponential, pspace, Where
X = Normal(0, 1, symbol=Symbol('x1'))
assert str(Where(X > 0)) == "Domain: 0 < x1"
D = Die(6, symbol=Symbol('d1'))
assert str(Where(D > 4)) == "Domain: Or(d1 == 5, d1 == 6)"
A = Exponential(1, symbol=Symbol('a'))
B = Exponential(1, symbol=Symbol('b'))
assert str(pspace(Tuple(A, B)).domain) == "Domain: And(0 <= a, 0 <= b)"
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == "Domain: 0 < x_{1}"
D = Die('d1', 6)
assert latex(where(D > 4)) == r"Domain: d_{1} = 5 \vee d_{1} = 6"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert latex(pspace(Tuple(A,
B)).domain) == "Domain: 0 \leq a \wedge 0 \leq b"
def test_symbolic_conditions():
B = Bernoulli('B', S(1) / 4)
D = Die('D', 4)
b, n = symbols('b, n')
Y = P(Eq(B, b))
Z = E(D > n)
assert Y == \
Piecewise((S(1)/4, Eq(b, 1)), (0, True)) + \
Piecewise((S(3)/4, Eq(b, 0)), (0, True))
assert Z == \
Piecewise((S(1)/4, n < 1), (0, True)) + Piecewise((S(1)/2, n < 2), (0, True)) + \
Piecewise((S(3)/4, n < 3), (0, True)) + Piecewise((S(1), n < 4), (0, True))
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_CDF():
D = Die(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_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))
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) == S(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)
assert variance(X + X) == 4 * variance(X)
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 P(X > 3) == S.Half
assert P(2 * X > 6) == S.Half
assert P(X > Y) == S(5) / 12
assert P(Eq(X, Y)) == P(Eq(X, 1))
assert E(X, X > 3) == 5
assert E(X, Y > 3) == E(X)
assert E(X + Y, Eq(X, Y)) == E(2 * X)
assert E(X + Y - Z, 2 * X > Y + 1) == S(49) / 12
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[S(22)] == S.One / 108 and d[S(4100)] == S.One / 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)
def test_sample_pymc3():
distribs_pymc3 = [Bernoulli('B', 0.2), Binomial('N', 5, 0.4)]
size = 3
pymc3 = import_module('pymc3')
if not pymc3:
skip('PyMC3 is not installed. Abort tests for _sample_pymc3.')
else:
for X in distribs_pymc3:
samps = sample(X, size=size, library='pymc3')
for sam in samps:
assert sam in X.pspace.domain.set
raises(NotImplementedError, lambda:
(sample(Die("D"), library='pymc3')))
def test_sample_pymc3():
distribs_pymc3 = [Bernoulli('B', 0.2), Binomial('N', 5, 0.4)]
size = 3
pymc3 = import_module('pymc3')
if not pymc3:
skip('PyMC3 is not installed. Abort tests for _sample_pymc3.')
else:
with ignore_warnings(
UserWarning
): ### TODO: Restore tests once warnings are removed
for X in distribs_pymc3:
samps = next(sample(X, size=size, library='pymc3'))
for sam in samps:
assert sam in X.pspace.domain.set
raises(NotImplementedError,
lambda: next(sample(Die("D"), library='pymc3')))
def test_Sample():
X = Die(6)
Y = Normal(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
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)
def test_discreteuniform():
# Symbolic
a, b, c = symbols('a b c')
X = DiscreteUniform('X', [a, b, c])
assert E(X) == (a + b + c) / 3
assert variance(X) == (a**2 + b**2 + c**2) / 3 - (a / 3 + b / 3 + c / 3)**2
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 skewness(Y) == 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 density(Die('D', 6)) == density(DiscreteUniform('U', range(1, 7)))
