def test_NormalDistribution():
nd = NormalDistribution(0, 1)
x = Symbol('x')
assert nd.cdf(x) == erf(sqrt(2) * x / 2) / 2 + S.Half
assert isinstance(nd.sample(), float) or nd.sample().is_Number
assert nd.expectation(1, x) == 1
assert nd.expectation(x, x) == 0
assert nd.expectation(x**2, x) == 1
#Test issue 10076
a = SingleContinuousPSpace(x, NormalDistribution(2, 4))
_z = Dummy('_z')
expected1 = Integral(
sqrt(2) * exp(-(_z - 2)**2 / 32) / (8 * sqrt(pi)), (_z, -oo, 1))
assert a.probability(x < 1, evaluate=False).dummy_eq(expected1) is True
expected2 = Integral(
sqrt(2) * exp(-(_z - 2)**2 / 32) / (8 * sqrt(pi)), (_z, 1, oo))
assert a.probability(x > 1, evaluate=False).dummy_eq(expected2) is True
b = SingleContinuousPSpace(x, NormalDistribution(1, 9))
expected3 = Integral(
sqrt(2) * exp(-(_z - 1)**2 / 162) / (18 * sqrt(pi)), (_z, 6, oo))
assert b.probability(x > 6, evaluate=False).dummy_eq(expected3) is True
expected4 = Integral(
sqrt(2) * exp(-(_z - 1)**2 / 162) / (18 * sqrt(pi)), (_z, -oo, 6))
assert b.probability(x < 6, evaluate=False).dummy_eq(expected4) is True
def probability(self, condition, **kwargs):
cond_inv = False
if isinstance(condition, Ne):
condition = Eq(condition.args[0], condition.args[1])
cond_inv = True
elif isinstance(condition, And): # they are independent
return Mul(*[self.probability(arg) for arg in condition.args])
elif isinstance(condition, Or): # they are independent
return Add(*[self.probability(arg) for arg in condition.args])
expr = condition.lhs - condition.rhs
rvs = random_symbols(expr)
dens = self.compute_density(expr)
if any([pspace(rv).is_Continuous for rv in rvs]):
from sympy.stats.crv import SingleContinuousPSpace
from sympy.stats.crv_types import ContinuousDistributionHandmade
if expr in self.values:
# Marginalize all other random symbols out of the density
randomsymbols = tuple(set(self.values) - frozenset([expr]))
symbols = tuple(rs.symbol for rs in randomsymbols)
pdf = self.domain.integrate(self.pdf, symbols, **kwargs)
return Lambda(expr.symbol, pdf)
dens = ContinuousDistributionHandmade(dens)
z = Dummy('z', real=True)
space = SingleContinuousPSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
else:
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import DiscreteDistributionHandmade
dens = DiscreteDistributionHandmade(dens)
z = Dummy('z', integer=True)
space = SingleDiscretePSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
return result if not cond_inv else S.One - result
def __new__(cls, sym, dist):
if isinstance(dist, SingleContinuousDistribution):
return SingleContinuousPSpace(sym, dist)
if isinstance(dist, SingleDiscreteDistribution):
return SingleDiscretePSpace(sym, dist)
sym = _symbol_converter(sym)
return Basic.__new__(cls, sym, dist)
def __new__(cls, sym, dist):
sym = sympify(sym)
if isinstance(dist, SingleContinuousDistribution):
return SingleContinuousPSpace(sym, dist)
if isinstance(dist, SingleDiscreteDistribution):
return SingleDiscretePSpace(sym, dist)
return Basic.__new__(cls, sym, dist)
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 pdf(self, *x):
dist = self.args[0]
z = Dummy('z')
if isinstance(dist, ContinuousDistribution):
rv = SingleContinuousPSpace(z, dist).value
elif isinstance(dist, DiscreteDistribution):
rv = SingleDiscretePSpace(z, dist).value
return MarginalDistribution(self, (rv,)).pdf(*x)
def __new__(cls, sym, dist):
if isinstance(dist, SingleContinuousDistribution):
return SingleContinuousPSpace(sym, dist)
if isinstance(dist, SingleDiscreteDistribution):
return SingleDiscretePSpace(sym, dist)
if isinstance(sym, str):
sym = Symbol(sym)
if not isinstance(sym, Symbol):
raise TypeError("s should have been string or Symbol")
return Basic.__new__(cls, sym, dist)
def __new__(cls, s, distribution):
s = _symbol_converter(s)
if isinstance(distribution, ContinuousDistribution):
return SingleContinuousPSpace(s, distribution)
if isinstance(distribution, DiscreteDistribution):
return SingleDiscretePSpace(s, distribution)
if isinstance(distribution, SingleFiniteDistribution):
return SingleFinitePSpace(s, distribution)
if not isinstance(distribution, CompoundDistribution):
raise ValueError("%s should be an isinstance of "
"CompoundDistribution"%(distribution))
return Basic.__new__(cls, s, distribution)
def test_ContinuousDistributionHandmade():
x = Symbol('x')
z = Dummy('z')
dens = Lambda(x, Piecewise((S.Half, (0<=x)&(x<1)), (0, (x>=1)&(x<2)),
(S.Half, (x>=2)&(x<3)), (0, True)))
dens = ContinuousDistributionHandmade(dens, set=Interval(0, 3))
space = SingleContinuousPSpace(z, dens)
assert dens.pdf == Lambda(x, Piecewise((1/2, (x >= 0) & (x < 1)),
(0, (x >= 1) & (x < 2)), (1/2, (x >= 2) & (x < 3)), (0, True)))
assert median(space.value) == Interval(1, 2)
assert E(space.value) == Rational(3, 2)
assert variance(space.value) == Rational(13, 12)
def _transform_pspace(self, sym, dist, pdf):
"""
This function returns the new pspace of the distribution using handmade
Distributions and their corresponding pspace.
"""
pdf = Lambda(sym, pdf(sym))
_set = dist.set
if isinstance(dist, ContinuousDistribution):
return SingleContinuousPSpace(sym, ContinuousDistributionHandmade(pdf, _set))
elif isinstance(dist, DiscreteDistribution):
return SingleDiscretePSpace(sym, DiscreteDistributionHandmade(pdf, _set))
elif isinstance(dist, SingleFiniteDistribution):
dens = dict((k, pdf(k)) for k in _set)
return SingleFinitePSpace(sym, FiniteDistributionHandmade(dens))