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Law of total variance

In probability theory, the law of total variance states that if X and Y are random variables on the same probability space, and the variance of X is finite, then

(The conditional expected value E( X | Y ) is a random variable in its own right, whose value depends on the value of Y. Notice that the conditional expected value of X given the event Y = y is a function of y (this is where adherence to the conventional rigidly case-sensitive notation of probability theory becomes important!). If we write E( X | Y = y) = g(y) then the random variable E( X | Y ) is just g(Y). Similar comments apply to the conditional variance.)

The nomenclature used here parallels the phrase law of total probability. See also law of total expectation.

A similar law for the third central moment μ3 says

Generalizations for higher moments than the third are messy; for higher cumulants on the other hand, a simple and elegant form exists.