from __future__ import print_function, division
from .rv import (probability, expectation, density, where, given, pspace, cdf,
sample, sample_iter, random_symbols, independent, dependent,
sampling_density)
from sympy import sqrt
__all__ = ['P', 'E', 'density', 'where', 'given', 'sample', 'cdf', 'pspace',
'sample_iter', 'variance', 'std', 'skewness', 'covariance',
'dependent', 'independent', 'random_symbols', 'correlation',
'moment', 'cmoment', 'sampling_density']
def moment(X, n, c=0, condition=None, **kwargs):
"""
Return the nth moment of a random expression about c i.e. E((X-c)**n)
Default value of c is 0.
Examples
========
>>> from sympy.stats import Die, moment, E
>>> X = Die('X', 6)
>>> moment(X, 1, 6)
-5/2
>>> moment(X, 2)
91/6
>>> moment(X, 1) == E(X)
True
"""
return expectation((X - c)**n, condition, **kwargs)
[docs]def variance(X, condition=None, **kwargs):
"""
Variance of a random expression
Expectation of (X-E(X))**2
Examples
========
>>> from sympy.stats import Die, E, Bernoulli, variance
>>> from sympy import simplify, Symbol
>>> X = Die('X', 6)
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)
>>> variance(2*X)
35/3
>>> simplify(variance(B))
p*(-p + 1)
"""
return cmoment(X, 2, condition, **kwargs)
def standard_deviation(X, condition=None, **kwargs):
"""
Standard Deviation of a random expression
Square root of the Expectation of (X-E(X))**2
Examples
========
>>> from sympy.stats import Bernoulli, std
>>> from sympy import Symbol, simplify
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)
>>> simplify(std(B))
sqrt(p*(-p + 1))
"""
return sqrt(variance(X, condition, **kwargs))
std = standard_deviation
[docs]def covariance(X, Y, condition=None, **kwargs):
"""
Covariance of two random expressions
The expectation that the two variables will rise and fall together
Covariance(X,Y) = E( (X-E(X)) * (Y-E(Y)) )
Examples
========
>>> from sympy.stats import Exponential, covariance
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> covariance(X, X)
lambda**(-2)
>>> covariance(X, Y)
0
>>> covariance(X, Y + rate*X)
1/lambda
"""
return expectation(
(X - expectation(X, condition, **kwargs)) *
(Y - expectation(Y, condition, **kwargs)),
condition, **kwargs)
def correlation(X, Y, condition=None, **kwargs):
"""
Correlation of two random expressions, also known as correlation
coefficient or Pearson's correlation
The normalized expectation that the two variables will rise
and fall together
Correlation(X,Y) = E( (X-E(X)) * (Y-E(Y)) / (sigma(X) * sigma(Y)) )
Examples
========
>>> from sympy.stats import Exponential, correlation
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> correlation(X, X)
1
>>> correlation(X, Y)
0
>>> correlation(X, Y + rate*X)
1/sqrt(1 + lambda**(-2))
"""
return covariance(X, Y, condition, **kwargs)/(std(X, condition, **kwargs)
* std(Y, condition, **kwargs))
def cmoment(X, n, condition=None, **kwargs):
"""
Return the nth central moment of a random expression about its mean
i.e. E((X - E(X))**n)
Examples
========
>>> from sympy.stats import Die, cmoment, variance
>>> X = Die('X', 6)
>>> cmoment(X, 3)
0
>>> cmoment(X, 2)
35/12
>>> cmoment(X, 2) == variance(X)
True
"""
mu = expectation(X, condition, **kwargs)
return moment(X, n, mu, condition, **kwargs)
def smoment(X, n, condition=None, **kwargs):
"""
Return the nth Standardized moment of a random expression i.e.
E( ((X - mu)/sigma(X))**n )
Examples
========
>>> from sympy.stats import skewness, Exponential, smoment
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> Y = Exponential('Y', rate)
>>> smoment(Y, 4)
9
>>> smoment(Y, 4) == smoment(3*Y, 4)
True
>>> smoment(Y, 3) == skewness(Y)
True
"""
sigma = std(X, condition, **kwargs)
return (1/sigma)**n*cmoment(X, n, condition, **kwargs)
def skewness(X, condition=None, **kwargs):
"""
Measure of the asymmetry of the probability distribution
Positive skew indicates that most of the values lie to the right of
the mean
skewness(X) = E( ((X - E(X))/sigma)**3 )
Examples
========
>>> from sympy.stats import skewness, Exponential, Normal
>>> from sympy import Symbol
>>> X = Normal('X', 0, 1)
>>> skewness(X)
0
>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> Y = Exponential('Y', rate)
>>> skewness(Y)
2
"""
return smoment(X, 3, condition, **kwargs)
P = probability
E = expectation
