Order isomorphism

In probability theory, the law of total covariance, covariance decomposition formula, or ECCE states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then

$\operatorname {cov} (X,Y)=\operatorname {E} (\operatorname {cov} (X,Y\mid Z))+\operatorname {cov} (\operatorname {E} (X\mid Z),\operatorname {E} (Y\mid Z)).\,$

The nomenclature in this article's title parallels the phrase law of total variance. Some writers on probability call this the "conditional covariance formula" or use other names.

(The conditional expected values E( X | Z ) and E( Y | Z ) are random variables in their own right, whose values depends on the value of Z. Notice that the conditional expected value of X given the event Z = z is a function of z (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is just g(Z). Similar comments apply to the conditional covariance.)

Proof

The law of total covariance can be proved using the law of total expectation: First,

$\operatorname {cov} [X,Y]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]$

from the definition of covariance. Then we apply the law of total expectation by conditioning on the random variable Z:

$=\operatorname {E} [\operatorname {E} [XY\mid Z]]-\operatorname {E} [\operatorname {E} [X\mid Z]]\operatorname {E} [\operatorname {E} [Y\mid Z]]$

Now we rewrite the term inside the first expectation using the definition of covariance:

$=\operatorname {E} \!\left[\operatorname {cov} [X,Y\mid Z]+\operatorname {E} [X\mid Z]\operatorname {E} [Y\mid Z]\right]-\operatorname {E} [\operatorname {E} [X\mid Z]]\operatorname {E} [\operatorname {E} [Y\mid Z]]$

Since expectation of a sum is the sum of expectations, we can regroup the terms:

$=\operatorname {E} \!\left[\operatorname {cov} [X,Y\mid Z]]+\operatorname {E} [\operatorname {E} [X\mid Z]\operatorname {E} [Y\mid Z]\right]-\operatorname {E} [\operatorname {E} [X\mid Z]]\operatorname {E} [\operatorname {E} [Y\mid Z]]$

Finally, we recognize the final two terms as the covariance of the conditional expectations E[X|Z] and E[Y|Z]:

$=\operatorname {E} (\operatorname {cov} (X,Y\mid Z))+\operatorname {cov} (\operatorname {E} (X\mid Z),\operatorname {E} (Y\mid Z))$

Notes and references

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See also

• Law of total variance, a special case corresponding to X = Y.