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 that the conditional expected value of X given the event Z = z is a function of z.
Similar comments apply to the conditional covariance.
The law of total covariance can be proved using the law of total expectation: First, from a simple standard identity on covariances.