The proposition in probability theory known as the law of total expectation,[1] the law of iterated expectations[2] (LIE), Adam's law,[3] the tower rule,[4] and the smoothing theorem,[5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then
i.e., the expected value of the conditional expected value of given is the same as the expected value of .
Note: The conditional expected value E(X | Y), with Y a random variable, is not a simple number; it is a random variable whose value depends on the value of Y. That is, the conditional expected value of X given the eventY = y is a number and it is a function of y. If we write g(y) for the value of E(X | Y = y) then the random variable E(X | Y) is g(Y).
One special case states that if is a finite or countable partition of the sample space, then
^Weiss, Neil A. (2005). A Course in Probability. Boston: Addison–Wesley. pp. 380–383. ISBN 0-321-18954-X.
^"Law of Iterated Expectation | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2018-03-28.
^"Adam's and Eve's Laws". Retrieved 2022-04-19.
^Rhee, Chang-han (Sep 20, 2011). "Probability and Statistics" (PDF).
^Wolpert, Robert (November 18, 2010). "Conditional Expectation" (PDF).
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