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
;
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.
References
[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) |
Comments
Almost-certain convergence is also called almost-sure convergence in the West.
References
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_mathematical_expectation&oldid=41628