// Workers AI · dad joke modeWhat did Law of total covariance say to its date? You vary much.

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 the law of total expectation: First,

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

from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z:

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

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

$=\operatorname {E} \!{\big [}\operatorname {cov} (X,Y\mid Z)+\operatorname {E} [X\mid Z]\operatorname {E} [Y\mid Z]{\big ]}-\operatorname {E} {\big [}\operatorname {E} [X\mid Z]{\big ]}\operatorname {E} {\big [}\operatorname {E} [Y\mid Z]{\big ]}$

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

$=\operatorname {E} \!{\big [}\operatorname {cov} (X,Y\mid Z){\big ]}+\operatorname {E} {\big [}\operatorname {E} [X\mid Z]\operatorname {E} [Y\mid Z]{\big ]}-\operatorname {E} {\big [}\operatorname {E} [X\mid Z]{\big ]}\operatorname {E} {\big [}\operatorname {E} [Y\mid Z]{\big ]}$

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

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

See also

• Law of total variance, a special case corresponding to X = Y.
• Law of total cumulance, of which the law of total covariance is a special case.