Conditional mathematical expectation

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conditional expectation, of a random variable

A function of an elementary event that characterizes the random variable with respect to a certain -algebra. Let be a probability space, let be a real-valued random variable with finite expectation defined on this space and let be a -algebra, . The conditional expectation of with respect to is understood to be a random variable , measurable with respect to and such that


for each . If the expectation of is infinite (but defined), i.e. only one of the numbers and is finite, then the definition of the conditional expectation by means of (*) still makes sense but may assume infinite values.

The conditional expectation is uniquely defined up to equivalence. In contrast to the mathematical expectation, which is a number, the conditional expectation represents a function (a random variable).

The properties of the conditional expectation are similar to those of the expectation:

1) if, almost certainly, ;

2) for every real ;

3) for arbitrary real and ;

4) ;

5) for every convex function . Furthermore, the following properties specific to the conditional expectation hold:

6) If is the trivial -algebra, then ;

7) ;

8) ;

9) if is independent of , then ;

10) if is measurable with respect to , then .

There is a theorem on convergence under the integral sign of conditional mathematical expectation: If is a sequence of random variables, , and almost certainly, then, almost certainly, .

The conditional expectation of a random variable with respect to a random variable is defined as the conditional expectation of relative to the -algebra generated by .

A particular case of the conditional expectation is the conditional probability.


[1] A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[3] J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970)
[4] M. Loève, "Probability theory" , Princeton Univ. Press (1963)


Almost-certain convergence is also called almost-sure convergence in the West.


[a1] R.B. Ash, "Real analysis and probability" , Acad. Press (1972)
[a2] J. Neveu, "Discrete-parameter martingales" , North-Holland (1975) (Translated from French)
How to Cite This Entry:
Conditional mathematical expectation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article