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 $\sigma$-algebra. Let $(\Omega, \mathcal{A}, \mathsf{P})$ be a probability space, let $X$ be a real-valued random variable with finite expectation defined on this space and let $\mathfrak{B}$ be a $\sigma$-algebra, $\mathfrak{B}\subseteq\mathcal{A}$. The conditional expectation of $X$ with respect to $\mathfrak{B}$ is understood to be a random variable $\mathsf{E}(X\, |\, \mathfrak{B})$, measurable with respect to $\mathfrak{B}$ and such that

$$\begin{equation}\tag{*} \int\limits_BX\mathsf{P}(d\,\omega)=\int\limits_B\mathsf{E}(X\, |\, \mathfrak{B})\mathsf{P}(d\,\omega) \end{equation}$$

for each $B\in\mathfrak{B}$. If the expectation of $X$ is infinite (but defined), i.e. only one of the numbers $\mathsf{E}X^+=\mathsf{E}\max(0, X)$ and $\mathsf{E}X^-=-\mathsf{E}\min(0, X)$ is finite, then the definition of the conditional expectation by means of (*) still makes sense but $\mathsf{E}(X\, |\, \mathfrak{B})$ 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) $\mathsf{E}(X_1\, |\, \mathfrak{B})\leq\mathsf{E}(X_2\, |\, \mathfrak{B})$ if, almost certainly, $X_1(\omega)\leq X_2(\omega)$;

2) $\mathsf{E}(c\, |\, \mathfrak{B})=c$ for every real $c$;

3) $\mathsf{E}(\alpha X_1+\beta X_2\, |\, \mathfrak{B})=\alpha\,\mathsf{E}(X_1\, |\, \mathfrak{B})+\beta\,\mathsf{E}(X_2\, |\, \mathfrak{B})$ for arbitrary real $\alpha$ and $\beta$;

4) $|\mathsf{E}(X\, |\, \mathfrak{B})|\leq\mathsf{E}(|X|\, |\, \mathfrak{B})$;

5) $g(\mathsf{E}(X\, |\, \mathfrak{B}))\leq\mathsf{E}(g(X)\, |\, \mathfrak{B})$ for every convex function $g$. Furthermore, the following properties specific to the conditional expectation hold:

6) If $\mathfrak{B}=\{\emptyset, \Omega\}$ is the trivial $\sigma$-algebra, then $\mathsf{E}(X\, |\, \mathfrak{B})=\mathsf{E}X$;

7) $\mathsf{E}(X\, |\, \mathcal{A})=X$;

8) $\mathsf{E}(\mathsf{E}(X\, |\, \mathfrak{B}))=\mathsf{E}X$;

9) if $X$ is independent of $\mathfrak{B}$, then $\mathsf{E}(X\, |\, \mathfrak{B})=\mathsf{E}X$;

10) if $Y$ is measurable with respect to $\mathfrak{B}$, then $\mathsf{E}(XY\, |\, \mathfrak{B})=Y\mathsf{E}(X\, |\, \mathfrak{B})$.

There is a theorem on convergence under the integral sign of conditional mathematical expectation: If $X_1, X_2, \dots$ is a sequence of random variables, $|X_n|\leq Y$, $n=1,2,\dots$ $\mathsf{E}Y<\infty$ and $X_n\rightarrow X$ almost certainly, then, almost certainly, $\mathsf{E}(X_n\, |\, \mathfrak{B})\rightarrow\mathsf{E}(X\, |\, \mathfrak{B})$.

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

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)