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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(more...)

        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, bounded = 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)
        


src/s/y/sympy-HEAD/sympy/stats/tests/test_continuous_rv.py   sympy(Download)
from sympy.stats import (P, E, where, density, variance, covariance, skewness,
                         given, pspace, cdf, ContinuousRV, sample,
                         Arcsin, Benini, Beta, BetaPrime, Cauchy,
                         Chi, ChiSquared,
                         ChiNoncentral, Dagum, Erlang, Exponential,
    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

src/s/y/sympy-HEAD/sympy/stats/tests/test_finite_rv.py   sympy(Download)
from sympy import (EmptySet, FiniteSet, S, Symbol, Interval, exp, erf, sqrt,
        symbols, simplify, Eq, cos, And, Tuple, Or, Dict, sympify, binomial,
        factor)
from sympy.stats import (DiscreteUniform, Die, Bernoulli, Coin, Binomial,
        Hypergeometric, P, E, variance, covariance, skewness, sample, density,
    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

src/s/y/sympy-HEAD/sympy/stats/tests/test_rv.py   sympy(Download)
from sympy import (EmptySet, FiniteSet, S, Symbol, Interval, exp, erf, sqrt,
        symbols, simplify, Eq, cos, And, Tuple, integrate, oo, sin, Sum, Basic,
        DiracDelta)
from sympy.stats import (Die, Normal, Exponential, P, E, variance, covariance,
        skewness, density, given, independent, dependent, where, pspace,