# Order isomorphism

In probability theory, the law of total covariance,[1] 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

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

The nomenclature in this article's title parallels the phrase law of total variance. Some writers on probability call this the "conditional covariance formula"[2] 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,

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

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

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

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

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

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

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

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

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

## Notes and references

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