# Source code for sympy.stats.crv

"""
Continuous Random Variables Module

========
sympy.stats.crv_types
sympy.stats.rv
sympy.stats.frv
"""

from __future__ import print_function, division

from sympy.stats.rv import (RandomDomain, SingleDomain, ConditionalDomain,
ProductDomain, PSpace, SinglePSpace, random_symbols, ProductPSpace,
NamedArgsMixin)
from sympy.functions.special.delta_functions import DiracDelta
from sympy import (Interval, Intersection, symbols, sympify, Dummy, Mul,
Integral, And, Or, Piecewise, cacheit, integrate, oo, Lambda,
Basic, S, exp, I, FiniteSet, Ne, Eq, Union)
from sympy.solvers.solveset import solveset
from sympy.solvers.inequalities import reduce_rational_inequalities
from sympy.polys.polyerrors import PolynomialError
import random

[docs]class ContinuousDomain(RandomDomain):
"""
A domain with continuous support

Represented using symbols and Intervals.
"""
is_Continuous = True

def as_boolean(self):
raise NotImplementedError("Not Implemented for generic Domains")

class SingleContinuousDomain(ContinuousDomain, SingleDomain):
"""
A univariate domain with continuous support

Represented using a single symbol and interval.
"""
def integrate(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
if not variables:
return expr
if frozenset(variables) != frozenset(self.symbols):
raise ValueError("Values should be equal")
# assumes only intervals
return Integral(expr, (self.symbol, self.set), **kwargs)

def as_boolean(self):
return self.set.as_relational(self.symbol)

class ProductContinuousDomain(ProductDomain, ContinuousDomain):
"""
A collection of independent domains with continuous support
"""

def integrate(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
for domain in self.domains:
domain_vars = frozenset(variables) & frozenset(domain.symbols)
if domain_vars:
expr = domain.integrate(expr, domain_vars, **kwargs)
return expr

def as_boolean(self):
return And(*[domain.as_boolean() for domain in self.domains])

class ConditionalContinuousDomain(ContinuousDomain, ConditionalDomain):
"""
A domain with continuous support that has been further restricted by a
condition such as x > 3
"""

def integrate(self, expr, variables=None, **kwargs):
if variables is None:
variables = self.symbols
if not variables:
return expr
# Extract the full integral
fullintgrl = self.fulldomain.integrate(expr, variables)
# separate into integrand and limits
integrand, limits = fullintgrl.function, list(fullintgrl.limits)

conditions = [self.condition]
while conditions:
cond = conditions.pop()
if cond.is_Boolean:
if isinstance(cond, And):
conditions.extend(cond.args)
elif isinstance(cond, Or):
raise NotImplementedError("Or not implemented here")
elif cond.is_Relational:
if cond.is_Equality:
# Add the appropriate Delta to the integrand
integrand *= DiracDelta(cond.lhs - cond.rhs)
else:
symbols = cond.free_symbols & set(self.symbols)
if len(symbols) != 1:  # Can't handle x > y
raise NotImplementedError(
"Multivariate Inequalities not yet implemented")
# Can handle x > 0
symbol = symbols.pop()
# Find the limit with x, such as (x, -oo, oo)
for i, limit in enumerate(limits):
if limit[0] == symbol:
# Make condition into an Interval like [0, oo]
cintvl = reduce_rational_inequalities_wrap(
cond, symbol)
# Make limit into an Interval like [-oo, oo]
lintvl = Interval(limit[1], limit[2])
# Intersect them to get [0, oo]
intvl = cintvl.intersect(lintvl)
# Put back into limits list
limits[i] = (symbol, intvl.left, intvl.right)
else:
raise TypeError(
"Condition %s is not a relational or Boolean" % cond)

return Integral(integrand, *limits, **kwargs)

def as_boolean(self):
return And(self.fulldomain.as_boolean(), self.condition)

@property
def set(self):
if len(self.symbols) == 1:
return (self.fulldomain.set & reduce_rational_inequalities_wrap(
self.condition, tuple(self.symbols)[0]))
else:
raise NotImplementedError(
"Set of Conditional Domain not Implemented")

class ContinuousDistribution(Basic):
def __call__(self, *args):
return self.pdf(*args)

class SingleContinuousDistribution(ContinuousDistribution, NamedArgsMixin):
""" Continuous distribution of a single variable

Serves as superclass for Normal/Exponential/UniformDistribution etc....

Represented by parameters for each of the specific classes.  E.g
NormalDistribution is represented by a mean and standard deviation.

Provides methods for pdf, cdf, and sampling

sympy.stats.crv_types.*
"""

set = Interval(-oo, oo)

def __new__(cls, *args):
args = list(map(sympify, args))
return Basic.__new__(cls, *args)

@staticmethod
def check(*args):
pass

def sample(self):
""" A random realization from the distribution """
icdf = self._inverse_cdf_expression()
return icdf(random.uniform(0, 1))

@cacheit
def _inverse_cdf_expression(self):
""" Inverse of the CDF

Used by sample
"""
x, z = symbols('x, z', real=True, positive=True, cls=Dummy)
# Invert CDF
try:
inverse_cdf = solveset(self.cdf(x) - z, x, S.Reals)
if isinstance(inverse_cdf, Intersection) and S.Reals in inverse_cdf.args:
inverse_cdf = list(inverse_cdf.args[1])
except NotImplementedError:
inverse_cdf = None
if not inverse_cdf or len(inverse_cdf) != 1:
raise NotImplementedError("Could not invert CDF")

return Lambda(z, inverse_cdf[0])

@cacheit
def compute_cdf(self, **kwargs):
""" Compute the CDF from the PDF

Returns a Lambda
"""
x, z = symbols('x, z', real=True, finite=True, cls=Dummy)
left_bound = self.set.start

# CDF is integral of PDF from left bound to z
pdf = self.pdf(x)
cdf = integrate(pdf, (x, left_bound, z), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((cdf, z >= left_bound), (0, True))
return Lambda(z, cdf)

def _cdf(self, x):
return None

def cdf(self, x, **kwargs):
""" Cumulative density function """
if len(kwargs) == 0:
cdf = self._cdf(x)
if cdf is not None:
return cdf
return self.compute_cdf(**kwargs)(x)

@cacheit
def compute_characteristic_function(self, **kwargs):
""" Compute the characteristic function from the PDF

Returns a Lambda
"""
x, t = symbols('x, t', real=True, finite=True, cls=Dummy)
pdf = self.pdf(x)
cf = integrate(exp(I*t*x)*pdf, (x, -oo, oo))
return Lambda(t, cf)

def _characteristic_function(self, t):
return None

def characteristic_function(self, t, **kwargs):
""" Characteristic function """
if len(kwargs) == 0:
cf = self._characteristic_function(t)
if cf is not None:
return cf
return self.compute_characteristic_function(**kwargs)(t)

def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
integral = Integral(expr * self.pdf(var), (var, self.set), **kwargs)
return integral.doit() if evaluate else integral

_argnames = ('pdf',)

@property
def set(self):
return self.args[1]

def __new__(cls, pdf, set=Interval(-oo, oo)):
return Basic.__new__(cls, pdf, set)

[docs]class ContinuousPSpace(PSpace):
""" Continuous Probability Space

Represents the likelihood of an event space defined over a continuum.

Represented with a ContinuousDomain and a PDF (Lambda-Like)
"""

is_Continuous = True
is_real = True

@property
def domain(self):
return self.args[0]

@property
def density(self):
return self.args[1]

@property
def pdf(self):
return self.density(*self.domain.symbols)

def integrate(self, expr, rvs=None, **kwargs):
if rvs is None:
rvs = self.values
else:
rvs = frozenset(rvs)

expr = expr.xreplace(dict((rv, rv.symbol) for rv in rvs))

domain_symbols = frozenset(rv.symbol for rv in rvs)

return self.domain.integrate(self.pdf * expr,
domain_symbols, **kwargs)

def compute_density(self, expr, **kwargs):
# Common case Density(X) where X in self.values
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)

z = Dummy('z', real=True, finite=True)
return Lambda(z, self.integrate(DiracDelta(expr - z), **kwargs))

@cacheit
def compute_cdf(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise ValueError(
"CDF not well defined on multivariate expressions")

d = self.compute_density(expr, **kwargs)
x, z = symbols('x, z', real=True, finite=True, cls=Dummy)
left_bound = self.domain.set.start

# CDF is integral of PDF from left bound to z
cdf = integrate(d(x), (x, left_bound, z), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((cdf, z >= left_bound), (0, True))
return Lambda(z, cdf)

@cacheit
def compute_characteristic_function(self, expr, **kwargs):
if not self.domain.set.is_Interval:
raise NotImplementedError("Characteristic function of multivariate expressions not implemented")

d = self.compute_density(expr, **kwargs)
x, t = symbols('x, t', real=True, cls=Dummy)
cf = integrate(exp(I*t*x)*d(x), (x, -oo, oo), **kwargs)
return Lambda(t, cf)

def probability(self, condition, **kwargs):
z = Dummy('z', real=True, finite=True)
cond_inv = False
if isinstance(condition, Ne):
condition = Eq(condition.args[0], condition.args[1])
cond_inv = True
# Univariate case can be handled by where
try:
domain = self.where(condition)
rv = [rv for rv in self.values if rv.symbol == domain.symbol][0]
# Integrate out all other random variables
pdf = self.compute_density(rv, **kwargs)
# return S.Zero if domain is empty set
if domain.set is S.EmptySet or isinstance(domain.set, FiniteSet):
return S.Zero if not cond_inv else S.One
if isinstance(domain.set, Union):
return sum(
Integral(pdf(z), (z, subset), **kwargs) for subset in
domain.set.args if isinstance(subset, Interval))
# Integrate out the last variable over the special domain
return Integral(pdf(z), (z, domain.set), **kwargs)

# Other cases can be turned into univariate case
# by computing a density handled by density computation
except NotImplementedError:
from sympy.stats.rv import density
expr = condition.lhs - condition.rhs
dens = density(expr, **kwargs)
if not isinstance(dens, ContinuousDistribution):
# Turn problem into univariate case
space = SingleContinuousPSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
return result if not cond_inv else S.One - result

def where(self, condition):
rvs = frozenset(random_symbols(condition))
if not (len(rvs) == 1 and rvs.issubset(self.values)):
raise NotImplementedError(
"Multiple continuous random variables not supported")
rv = tuple(rvs)[0]
interval = reduce_rational_inequalities_wrap(condition, rv)
interval = interval.intersect(self.domain.set)
return SingleContinuousDomain(rv.symbol, interval)

def conditional_space(self, condition, normalize=True, **kwargs):
condition = condition.xreplace(dict((rv, rv.symbol) for rv in self.values))
domain = ConditionalContinuousDomain(self.domain, condition)
if normalize:
# create a clone of the variable to
# make sure that variables in nested integrals are different
# from the variables outside the integral
# this makes sure that they are evaluated separately
# and in the correct order
replacement  = {rv: Dummy(str(rv)) for rv in self.symbols}
norm = domain.integrate(self.pdf, **kwargs)
pdf = self.pdf / norm.xreplace(replacement)
density = Lambda(domain.symbols, pdf)

return ContinuousPSpace(domain, density)

class SingleContinuousPSpace(ContinuousPSpace, SinglePSpace):
"""
A continuous probability space over a single univariate variable

These consist of a Symbol and a SingleContinuousDistribution

This class is normally accessed through the various random variable
functions, Normal, Exponential, Uniform, etc....
"""

@property
def set(self):
return self.distribution.set

@property
def domain(self):
return SingleContinuousDomain(sympify(self.symbol), self.set)

def sample(self):
"""
Internal sample method

Returns dictionary mapping RandomSymbol to realization value.
"""
return {self.value: self.distribution.sample()}

def integrate(self, expr, rvs=None, **kwargs):
rvs = rvs or (self.value,)
if self.value not in rvs:
return expr

expr = expr.xreplace(dict((rv, rv.symbol) for rv in rvs))

x = self.value.symbol
try:
return self.distribution.expectation(expr, x, evaluate=False, **kwargs)
except Exception:
return Integral(expr * self.pdf, (x, self.set), **kwargs)

def compute_cdf(self, expr, **kwargs):
if expr == self.value:
z = symbols("z", real=True, finite=True, cls=Dummy)
return Lambda(z, self.distribution.cdf(z, **kwargs))
else:
return ContinuousPSpace.compute_cdf(self, expr, **kwargs)

def compute_characteristic_function(self, expr, **kwargs):
if expr == self.value:
t = symbols("t", real=True, cls=Dummy)
return Lambda(t, self.distribution.characteristic_function(t, **kwargs))
else:
return ContinuousPSpace.compute_characteristic_function(self, expr, **kwargs)

def compute_density(self, expr, **kwargs):
# http://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables
if expr == self.value:
return self.density
y = Dummy('y')

gs = solveset(expr - y, self.value, S.Reals)

if isinstance(gs, Intersection) and S.Reals in gs.args:
gs = list(gs.args[1])

if not gs:
raise ValueError("Can not solve %s for %s"%(expr, self.value))
fx = self.compute_density(self.value)
fy = sum(fx(g) * abs(g.diff(y)) for g in gs)
return Lambda(y, fy)

class ProductContinuousPSpace(ProductPSpace, ContinuousPSpace):
"""
A collection of independent continuous probability spaces
"""
@property
def pdf(self):
p = Mul(*[space.pdf for space in self.spaces])
return p.subs(dict((rv, rv.symbol) for rv in self.values))

def _reduce_inequalities(conditions, var, **kwargs):
try:
return reduce_rational_inequalities(conditions, var, **kwargs)
except PolynomialError:
raise ValueError("Reduction of condition failed %s\n" % conditions[0])

def reduce_rational_inequalities_wrap(condition, var):
if condition.is_Relational:
return _reduce_inequalities([[condition]], var, relational=False)
if condition.__class__ is Or:
return _reduce_inequalities([list(condition.args)],
var, relational=False)
if condition.__class__ is And:
intervals = [_reduce_inequalities([[arg]], var, relational=False)
for arg in condition.args]
I = intervals[0]
for i in intervals:
I = I.intersect(i)
return I