def test_binomial_quantile():
X = Binomial('X', 50, S.Half)
assert quantile(X)(0.95) == S(31)
X = Binomial('X', 5, S.Half)
p = Symbol("p", positive=True)
assert quantile(X)(p) == Piecewise((nan, p > S.One), (S.Zero, p <= Rational(1, 32)),\
(S.One, p <= Rational(3, 16)), (S(2), p <= S.Half), (S(3), p <= Rational(13, 16)),\
(S(4), p <= Rational(31, 32)), (S(5), p <= S.One))
def test_binomial_quantile():
X = Binomial('X', 50, S.Half)
assert quantile(X)(0.95) == S(31)
X = Binomial('X', 5, S(1) / 2)
p = Symbol("p", positive=True)
assert quantile(X)(p) == Piecewise((nan, p > S(1)), (S(0), p <= S(1)/32),\
(S(1), p <= S(3)/16), (S(2), p <= S(1)/2), (S(3), p <= S(13)/16),\
(S(4), p <= S(31)/32), (S(5), p <= S(1)))
def test_binomial_quantile():
X = Binomial('X', 50, S.Half)
assert quantile(X)(0.95) == S(31)
X = Binomial('X', 5, S(1)/2)
p = Symbol("p", positive=True)
assert quantile(X)(p) == Piecewise((nan, p > S(1)), (S(0), p <= S(1)/32),\
(S(1), p <= S(3)/16), (S(2), p <= S(1)/2), (S(3), p <= S(13)/16),\
(S(4), p <= S(31)/32), (S(5), p <= S(1)))
def test_exponential():
rate = Symbol('lambda', positive=True)
X = Exponential('x', rate)
p = Symbol("p", positive=True, real=True, finite=True)
assert E(X) == 1 / rate
assert variance(X) == 1 / rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert kurtosis(X) == 9
assert kurtosis(X) == smoment(X, 4)
assert smoment(2 * X, 4) == smoment(X, 4)
assert moment(X, 3) == 3 * 2 * 1 / rate**3
assert P(X > 0) is S.One
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10 * rate)
assert quantile(X)(p) == -log(1 - p) / rate
assert where(X <= 1).set == Interval(0, 1)
#Test issue 9970
z = Dummy('z')
assert P(X > z) == exp(-z * rate)
assert P(X < z) == 0
#Test issue 10076 (Distribution with interval(0,oo))
x = Symbol('x')
_z = Dummy('_z')
b = SingleContinuousPSpace(x, ExponentialDistribution(2))
expected1 = Integral(2 * exp(-2 * _z), (_z, 3, oo))
assert b.probability(x > 3, evaluate=False).dummy_eq(expected1) is True
expected2 = Integral(2 * exp(-2 * _z), (_z, 0, 4))
assert b.probability(x < 4, evaluate=False).dummy_eq(expected2) is True
def test_loglogistic():
a, b = symbols('a b')
assert LogLogistic('x', a, b)
a = Symbol('a', negative=True)
b = Symbol('b', positive=True)
raises(ValueError, lambda: LogLogistic('x', a, b))
a = Symbol('a', positive=True)
b = Symbol('b', negative=True)
raises(ValueError, lambda: LogLogistic('x', a, b))
a, b, z, p = symbols('a b z p', positive=True)
X = LogLogistic('x', a, b)
assert density(X)(z) == b*(z/a)**(b - 1)/(a*((z/a)**b + 1)**2)
assert cdf(X)(z) == 1/(1 + (z/a)**(-b))
assert quantile(X)(p) == a*(p/(1 - p))**(1/b)
# Expectation
assert E(X) == Piecewise((S.NaN, b <= 1), (pi*a/(b*sin(pi/b)), True))
b = symbols('b', prime=True) # b > 1
X = LogLogistic('x', a, b)
assert E(X) == pi*a/(b*sin(pi/b))
X = LogLogistic('x', 1, 2)
assert median(X) == FiniteSet(1)
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_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_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_bernoulli():
p, a, b, t = symbols("p a b t")
X = Bernoulli("B", p, a, b)
assert E(X) == a * p + b * (-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
assert characteristic_function(X)(
t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t)
assert moment_generating_function(X)(
t) == p * exp(a * t) + (-p + 1) * exp(b * t)
X = Bernoulli("B", p, 1, 0)
z = Symbol("z")
assert E(X) == p
assert simplify(variance(X)) == p * (1 - p)
assert E(a * X + b) == a * E(X) + b
assert simplify(variance(a * X + b)) == simplify(a**2 * variance(X))
assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)),
(0, z <= 1 - p), (1, z <= 1))
raises(ValueError, lambda: Bernoulli("B", 1.5))
raises(ValueError, lambda: Bernoulli("B", -0.5))
# issue 8248
assert X.pspace.compute_expectation(1) == 1
def test_logistic():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
p = Symbol("p", positive=True)
X = Logistic('x', mu, s)
assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1)
assert quantile(X)(p) == mu - s*log(-S.One + 1/p)
def test_logistic():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
p = Symbol("p", positive=True)
X = Logistic('x', mu, s)
assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1)
assert quantile(X)(p) == mu - s*log(-S(1) + 1/p)
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_cauchy():
x0 = Symbol("x0")
gamma = Symbol("gamma", positive=True)
p = Symbol("p", positive=True)
X = Cauchy('x', x0, gamma)
assert density(X)(x) == 1 / (pi * gamma * (1 + (x - x0)**2 / gamma**2))
assert diff(cdf(X)(x), x) == density(X)(x)
assert quantile(X)(p) == gamma * tan(pi * (p - S.Half)) + x0
gamma = Symbol("gamma", nonpositive=True)
raises(ValueError, lambda: Cauchy('x', x0, gamma))
def test_cauchy():
x0 = Symbol("x0")
gamma = Symbol("gamma", positive=True)
p = Symbol("p", positive=True)
X = Cauchy('x', x0, gamma)
assert density(X)(x) == 1/(pi*gamma*(1 + (x - x0)**2/gamma**2))
assert diff(cdf(X)(x), x) == density(X)(x)
assert quantile(X)(p) == gamma*tan(pi*(p - S.Half)) + x0
gamma = Symbol("gamma", positive=False)
raises(ValueError, lambda: Cauchy('x', x0, gamma))
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_logistic():
mu = Symbol("mu", real=True)
s = Symbol("s", positive=True)
p = Symbol("p", positive=True)
X = Logistic('x', mu, s)
#Tests characteristics_function
assert characteristic_function(X)(x) == \
(Piecewise((pi*s*x*exp(I*mu*x)/sinh(pi*s*x), Ne(x, 0)), (1, True)))
assert density(X)(x) == exp((-x + mu)/s)/(s*(exp((-x + mu)/s) + 1)**2)
assert cdf(X)(x) == 1/(exp((mu - x)/s) + 1)
assert quantile(X)(p) == mu - s*log(-S.One + 1/p)
def test_cauchy():
x0 = Symbol("x0")
gamma = Symbol("gamma", positive=True)
p = Symbol("p", positive=True)
X = Cauchy('x', x0, gamma)
# Tests the characteristic function
assert characteristic_function(X)(x) == exp(-gamma * Abs(x) + I * x * x0)
assert density(X)(x) == 1 / (pi * gamma * (1 + (x - x0)**2 / gamma**2))
assert diff(cdf(X)(x), x) == density(X)(x)
assert quantile(X)(p) == gamma * tan(pi * (p - S.Half)) + x0
gamma = Symbol("gamma", nonpositive=True)
raises(ValueError, lambda: Cauchy('x', x0, gamma))
def test_single_normal():
mu = Symbol('mu', real=True, finite=True)
sigma = Symbol('sigma', real=True, positive=True, finite=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert simplify(E(Y)) == mu
assert simplify(variance(Y)) == sigma**2
pdf = density(Y)
x = Symbol('x')
assert (pdf(x) ==
2**S.Half*exp(-(mu - x)**2/(2*sigma**2))/(2*pi**S.Half*sigma))
assert P(X**2 < 1) == erf(2**S.Half/2)
assert quantile(Y)(x) == Intersection(S.Reals, FiniteSet(sqrt(2)*sigma*(sqrt(2)*mu/(2*sigma) + erfinv(2*x - 1))))
assert E(X, Eq(X, mu)) == mu
def test_single_normal():
mu = Symbol('mu', real=True)
sigma = Symbol('sigma', positive=True)
X = Normal('x', 0, 1)
Y = X*sigma + mu
assert E(Y) == mu
assert variance(Y) == sigma**2
pdf = density(Y)
x = Symbol('x', real=True)
assert (pdf(x) ==
2**S.Half*exp(-(x - mu)**2/(2*sigma**2))/(2*pi**S.Half*sigma))
assert P(X**2 < 1) == erf(2**S.Half/2)
assert quantile(Y)(x) == Intersection(S.Reals, FiniteSet(sqrt(2)*sigma*(sqrt(2)*mu/(2*sigma) + erfinv(2*x - 1))))
assert E(X, Eq(X, mu)) == mu
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_exponential():
rate = Symbol('lambda', positive=True, real=True, finite=True)
X = Exponential('x', rate)
p = Symbol("p", positive=True, real=True,finite=True)
assert E(X) == 1/rate
assert variance(X) == 1/rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert smoment(2*X, 4) == smoment(X, 4)
assert moment(X, 3) == 3*2*1/rate**3
assert P(X > 0) == S(1)
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10*rate)
assert quantile(X)(p) == -log(1-p)/rate
assert where(X <= 1).set == Interval(0, 1)
def test_exponential():
rate = Symbol('lambda', positive=True)
X = Exponential('x', rate)
p = Symbol("p", positive=True, real=True, finite=True)
assert E(X) == 1 / rate
assert variance(X) == 1 / rate**2
assert skewness(X) == 2
assert skewness(X) == smoment(X, 3)
assert smoment(2 * X, 4) == smoment(X, 4)
assert moment(X, 3) == 3 * 2 * 1 / rate**3
assert P(X > 0) == S(1)
assert P(X > 1) == exp(-rate)
assert P(X > 10) == exp(-10 * rate)
assert quantile(X)(p) == -log(1 - p) / rate
assert where(X <= 1).set == Interval(0, 1)
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_bernoulli():
p, a, b, t = symbols('p a b t')
X = Bernoulli('B', p, a, b)
assert E(X) == a*p + b*(-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
assert characteristic_function(X)(t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t)
assert moment_generating_function(X)(t) == p * exp(a * t) + (-p + 1) * exp(b * t)
X = Bernoulli('B', p, 1, 0)
z = Symbol("z")
assert E(X) == p
assert simplify(variance(X)) == p*(1 - p)
assert E(a*X + b) == a*E(X) + b
assert simplify(variance(a*X + b)) == simplify(a**2 * variance(X))
assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)), (0, z <= 1 - p), (1, z <= 1))
raises(ValueError, lambda: Bernoulli('B', 1.5))
raises(ValueError, lambda: Bernoulli('B', -0.5))
def test_bernoulli():
p, a, b, t = symbols('p a b t')
X = Bernoulli('B', p, a, b)
assert E(X) == a * p + b * (-p + 1)
assert density(X)[a] == p
assert density(X)[b] == 1 - p
assert characteristic_function(X)(
t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t)
assert moment_generating_function(X)(
t) == p * exp(a * t) + (-p + 1) * exp(b * t)
X = Bernoulli('B', p, 1, 0)
z = Symbol("z")
assert E(X) == p
assert simplify(variance(X)) == p * (1 - p)
assert E(a * X + b) == a * E(X) + b
assert simplify(variance(a * X + b)) == simplify(a**2 * variance(X))
assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)),
(0, z <= 1 - p), (1, z <= 1))
Y = Bernoulli('Y', Rational(1, 2))
assert median(Y) == FiniteSet(0, 1)
Z = Bernoulli('Z', Rational(2, 3))
assert median(Z) == FiniteSet(1)
raises(ValueError, lambda: Bernoulli('B', 1.5))
raises(ValueError, lambda: Bernoulli('B', -0.5))
#issue 8248
assert X.pspace.compute_expectation(1) == 1
p = Rational(1, 5)
X = Binomial('X', 5, p)
Y = Binomial('Y', 7, 2 * p)
Z = Binomial('Z', 9, 3 * p)
assert coskewness(Y + Z, X + Y, X + Z).simplify() == 0
assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y).simplify() == \
sqrt(1529)*Rational(12, 16819)
assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y, X < 2).simplify() \
== -sqrt(357451121)*Rational(2812, 4646864573)
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