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
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_pspace():
X, Y = Normal(0,1), Normal(0,1)
assert not pspace(5+3)
assert pspace(X) == X.pspace
assert pspace(2*X+1) == X.pspace
assert pspace(2*X+Y) == ProductPSpace(Y.pspace, X.pspace)
def test_domains():
x, y = symbols('x y')
X, Y= Die(6, symbol=x), Die(6, symbol=y)
# 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(4, symbol=x)
raises(ValueError, "P(X>Z)") # Two domains with same internal symbol
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_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)
def test_pspace():
X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
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) == ProductPSpace(Y.pspace, X.pspace)
def test_FiniteRV():
F = FiniteRV("F", {1: S.Half, 2: S.One / 4, 3: S.One / 4})
assert dict(density(F).items()) == {S(1): S.Half, S(2): S.One / 4, S(3): S.One / 4}
assert P(F >= 2) == S.Half
assert pspace(F).domain.as_boolean() == Or(*[Eq(F.symbol, i) for i in [1, 2, 3]])
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) == 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 quantile(X)(p) == Piecewise((nan, (p > S.One) | (p < S(0))),\
(S.One, p <= S(1)/6), (S(2), p <= S(1)/3), (S(3), p <= S.Half),\
(S(4), p <= S(2)/3), (S(5), p <= S(5)/6), (S(6), p <= S.One))
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_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_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_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_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('x1', 0, 1)
assert str(where(X > 0)) == "Domain: 0 < x1"
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(0 <= a, 0 <= b)"
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal(0, 1, symbol=Symbol('x1'))
assert latex(where(X>0)) == "Domain: 0 < x_{1}"
D = Die(6, symbol=Symbol('d1'))
assert latex(where(D>4)) == r"Domain: d_{1} = 5 \vee d_{1} = 6"
A = Exponential(1, symbol=Symbol('a'))
B = Exponential(1, symbol=Symbol('b'))
assert latex(pspace(Tuple(A,B)).domain) =="Domain: 0 \leq a \wedge 0 \leq b"
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_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == "Domain: x_{1} > 0"
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: a \geq 0 \wedge b \geq 0"
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_FiniteRV():
F = FiniteRV('F', {1: S.Half, 2: S.One / 4, 3: S.One / 4})
assert dict(density(F).items()) == {
S(1): S.Half,
S(2): S.One / 4,
S(3): S.One / 4
}
assert P(F >= 2) == S.Half
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
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_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_FiniteRV():
F = FiniteRV('F', {1: S.Half, 2: S.One/4, 3: S.One/4})
assert dict(density(F).items()) == {S(1): S.Half, S(2): S.One/4, S(3): S.One/4}
assert P(F >= 2) == S.Half
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S.Half, 3: S.Half}))
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S(-1)/2, 3: S.One}))
raises(ValueError, lambda: FiniteRV('F', {1: S.One, 2: S(3)/2, 3: S.Zero, 4: S(-1)/2, 5: S(-3)/4, 6: S(-1)/4}))
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) == 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 quantile(X)(p) == Piecewise((nan, (p > S.One) | (p < S(0))),\
(S.One, p <= S(1)/6), (S(2), p <= S(1)/3), (S(3), p <= S.Half),\
(S(4), p <= S(2)/3), (S(5), p <= S(5)/6), (S(6), p <= S.One))
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_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): S.One / 4, (H, T): S.One / 4, (T, H): S.One / 4, (T, T): S.One / 4}
assert dict(density(C).items()) == {H: S.Half, T: S.Half}
F = Coin("F", S.One / 10)
assert P(Eq(F, H)) == S(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_coins():
C, D = Coin(), Coin()
H, T = sorted(density(C).keys())
assert P(Eq(C, D)) == S.Half
assert density(Tuple(C, D)) == {(H, H): S.One / 4, (H, T): S.One / 4, (T, H): S.One / 4, (T, T): S.One / 4}
assert density(C) == {H: S.Half, T: S.Half}
E = Coin(S.One / 10)
assert P(Eq(E, H)) == S(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_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): S.One/4, (H, T): S.One/4,
(T, H): S.One/4, (T, T): S.One/4}
assert dict(density(C).items()) == {H: S.Half, T: S.Half}
F = Coin('F', S.One/10)
assert P(Eq(F, H)) == S(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_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
assert characteristic_function(B)(x) == hyper((a,), (a + b,), I*x)
assert density(B)(x) == x**(a - 1)*(1 - x)**(b - 1)/beta(a, b)
assert simplify(E(B)) == a / (a + b)
assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2)
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta('x', a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
def test_FiniteRV():
F = FiniteRV('F', {1: S.Half, 2: S.One/4, 3: S.One/4})
p = Symbol("p", positive=True)
assert dict(density(F).items()) == {S(1): S.Half, S(2): S.One/4, S(3): S.One/4}
assert P(F >= 2) == S.Half
assert quantile(F)(p) == Piecewise((nan, p > S.One), (S.One, p <= S.Half),\
(S(2), p <= S(3)/4),(S(3), True))
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S.Half, 3: S.Half}))
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S(-1)/2, 3: S.One}))
raises(ValueError, lambda: FiniteRV('F', {1: S.One, 2: S(3)/2, 3: S.Zero,\
4: S(-1)/2, 5: S(-3)/4, 6: S(-1)/4}))
def test_FiniteRV():
F = FiniteRV('F', {1: S.Half, 2: S.One/4, 3: S.One/4})
p = Symbol("p", positive=True)
assert dict(density(F).items()) == {S(1): S.Half, S(2): S.One/4, S(3): S.One/4}
assert P(F >= 2) == S.Half
assert quantile(F)(p) == Piecewise((nan, p > S.One), (S.One, p <= S.Half),\
(S(2), p <= S(3)/4),(S(3), True))
assert pspace(F).domain.as_boolean() == Or(
*[Eq(F.symbol, i) for i in [1, 2, 3]])
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S.Half, 3: S.Half}))
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: S(-1)/2, 3: S.One}))
raises(ValueError, lambda: FiniteRV('F', {1: S.One, 2: S(3)/2, 3: S.Zero,\
4: S(-1)/2, 5: S(-3)/4, 6: S(-1)/4}))
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) == 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)
def test_FiniteRV():
F = FiniteRV('F', {1: S.Half, 2: Rational(1, 4), 3: Rational(1, 4)})
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}))
raises(ValueError, lambda: FiniteRV('F', {1: S.Half, 2: Rational(-1, 2), 3: S.One}))
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)}))
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) == 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)
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_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta(a, b)
assert pspace(B).domain.set == Interval(0, 1)
x, dens = Density(B)
assert dens == x**(a-1)*(1-x)**(b-1) / beta(a,b)
# This is too slow
# assert E(B) == a / (a + b)
# assert Var(B) == (a*b) / ((a+b)**2 * (a+b+1))
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta(a, b)
assert E(B) == a / S(a + b)
assert Var(B) == (a*b) / S((a+b)**2 * (a+b+1))
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
assert simplify(E(B)) == a / (a + b)
assert simplify(variance(B)) == a*b / (a**3 + 3*a**2*b + a**2 + 3*a*b**2 + 2*a*b + b**3 + b**2)
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta('x', a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
def test_beta():
a, b = symbols('alpha beta', positive=True)
B = Beta('x', a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol('x')
assert dens(x) == x**(a - 1)*(1 - x)**(b - 1) / beta(a, b)
# This is too slow
# assert E(B) == a / (a + b)
# assert variance(B) == (a*b) / ((a+b)**2 * (a+b+1))
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta('x', a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a*b) / S((a + b)**2 * (a + b + 1))
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_coins():
C, D = Coin(), Coin()
H, T = sorted(Density(C).keys())
assert P(Eq(C, D)) == S.Half
assert Density(Tuple(C, D)) == {
(H, H): S.One / 4,
(H, T): S.One / 4,
(T, H): S.One / 4,
(T, T): S.One / 4
}
assert Density(C) == {H: S.Half, T: S.Half}
E = Coin(S.One / 10)
assert P(Eq(E, H)) == S(1) / 10
d = pspace(C).domain
assert d.as_boolean() == Or(Eq(C.symbol, H), Eq(C.symbol, T))
raises(ValueError, "P(C>D)") # Can't intelligently compare H to T
def test_beta():
a, b = symbols("alpha beta", positive=True)
B = Beta("x", a, b)
assert pspace(B).domain.set == Interval(0, 1)
dens = density(B)
x = Symbol("x")
assert dens(x) == x ** (a - 1) * (1 - x) ** (b - 1) / beta(a, b)
# This is too slow
# assert E(B) == a / (a + b)
# assert variance(B) == (a*b) / ((a+b)**2 * (a+b+1))
# Full symbolic solution is too much, test with numeric version
a, b = 1, 2
B = Beta("x", a, b)
assert expand_func(E(B)) == a / S(a + b)
assert expand_func(variance(B)) == (a * b) / S((a + b) ** 2 * (a + b + 1))
def test_dice():
X, Y, Z = Die(6), Die(6), Die(6)
a, b = symbols('a b')
assert E(X) == 3 + S.Half
assert Var(X) == S(35) / 12
assert E(X + Y) == 7
assert E(X + X) == 7
assert E(a * X + b) == a * E(X) + b
assert Var(X + Y) == Var(X) + Var(Y)
assert Var(X + X) == 4 * Var(X)
assert Covar(X, Y) == S.Zero
assert Covar(X, X + Y) == Var(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_dice():
X, Y, Z= Die(6), Die(6), Die(6)
a,b = symbols('a b')
assert E(X) == 3+S.Half
assert Var(X) == S(35)/12
assert E(X+Y) == 7
assert E(X+X) == 7
assert E(a*X+b) == a*E(X)+b
assert Var(X+Y) == Var(X) + Var(Y)
assert Var(X+X) == 4 * Var(X)
assert Covar(X,Y) == S.Zero
assert Covar(X, X+Y) == Var(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_random_parameters():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert density(meas, evaluate=False)(z)
assert isinstance(pspace(meas), IndependentProductPSpace)
def test_random_parameters():
mu = Normal('mu', 2, 3)
meas = Normal('T', mu, 1)
assert density(meas, evaluate=False)(z)
assert isinstance(pspace(meas), ProductPSpace)
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) == 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 + 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) == 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 quantile(X)(p) == Piecewise((nan, (p > S.One) | (p < S(0))),\
(S.One, p <= S(1)/6), (S(2), p <= S(1)/3), (S(3), p <= S.Half),\
(S(4), p <= S(2)/3), (S(5), p <= S(5)/6), (S(6), p <= S.One))
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
# 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()) == set([1, 2, 3, 4])
assert set(dens.subs(n, 4).doit().values()) == set([S(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() == S(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() == S(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_FinitePSpace():
X = Die('X', 6)
space = pspace(X)
assert space.density == DieDistribution(6)
def test_FinitePSpace():
X = Die('X', 6)
space = pspace(X)
assert space.density == DieDistribution(6)