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. N.G. Ushakov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Conditional_mathematical_expectation&oldid=15801