In probability theory, the law of total covariance,[1] 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
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.
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
[edit]The law of total covariance can be proved using the law of total expectation: First,
from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z:
Now we rewrite the term inside the first expectation using the definition of covariance:
Since expectation of a sum is the sum of expectations, we can regroup the terms:
Finally, we recognize the final two terms as the covariance of the conditional expectations E[X | Z] and E[Y | Z]:
See also
[edit]- 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.