def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
p = Symbol("p", positive=True)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2 * X + Y, -X) == -2 * variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert kurtosis(X) == 3
assert kurtosis(X + Y) == 3
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X * X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert smoment(X + Y, 4) == kurtosis(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
assert quantile(X)(p) == sqrt(2) * erfinv(2 * p - S.One)
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_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 symstat():
"""
check out https://docs.sympy.org/latest/modules/stats.html
"""
from sympy.stats import E
X, Z = Die('X', 6), Die('Z', 6)
Y = X + Z
print('E[X]: {}'.format(E(X)))
print('\n')
print('E[Y]: {}'.format(E(Y)))
print('\n')
for x in list(density(X)):
print('E[Y|X = {0}] = {1}'.format(x, E(Y, X=x))) # this conditioning isn't quite right
print('\n')
for y in list(density(Y)):
print('E[X|Y = {0}] = {1}'.format(y, E(X, Y=y)))
print('\n')
print('Var(X): {}'.format(variance(X)))
print('\n')
print('Var(Y): {}'.format(variance(Y)))
print('\n')
print('Cov(X, Y): {}'.format(covariance(X, X + Y)))
print('\n')
print('rho(X, Y): {}'.format(correlation(X, X + Y)))
print('\n')
print('Cov(X, Z): {}'.format(covariance(X, Y)))
print('\n')
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_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_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
def test_multiple_normal():
X, Y = Normal('x', 0, 1), Normal('y', 0, 1)
p = Symbol("p", positive=True)
assert E(X + Y) == 0
assert variance(X + Y) == 2
assert variance(X + X) == 4
assert covariance(X, Y) == 0
assert covariance(2*X + Y, -X) == -2*variance(X)
assert skewness(X) == 0
assert skewness(X + Y) == 0
assert correlation(X, Y) == 0
assert correlation(X, X + Y) == correlation(X, X - Y)
assert moment(X, 2) == 1
assert cmoment(X, 3) == 0
assert moment(X + Y, 4) == 12
assert cmoment(X, 2) == variance(X)
assert smoment(X*X, 2) == 1
assert smoment(X + Y, 3) == skewness(X + Y)
assert E(X, Eq(X + Y, 0)) == 0
assert variance(X, Eq(X + Y, 0)) == S.Half
assert quantile(X)(p) == sqrt(2)*erfinv(2*p - S.One)
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