Conditional mathematical expectation
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 ;
5) for every convex function . Furthermore, the following properties specific to the conditional expectation hold:
6) If is the trivial -algebra, then ;
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.
|||A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)|
|||Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)|
|||J. Neveu, "Bases mathématiques du calcul des probabilités" , Masson (1970)|
|||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)|
Conditional mathematical expectation. N.G. Ushakov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Conditional_mathematical_expectation&oldid=15801