# 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 μ_{3} says

Generalizations for higher moments than the third are messy; for higher

cumulants on the other hand, a simple and elegant form exists.