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Concept# Law of total covariance

Summary

In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula 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.
Note: The conditional expected values E( X | Z ) and E( Y | Z ) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is g(Z). Similar comments apply to the conditional covariance.
Proof
The law of total covariance can be proved using

Official source

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