View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245002.png" />-algebra. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245003.png" /> be a probability space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245004.png" /> be a real-valued random variable with finite expectation defined on this space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245005.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245006.png" />-algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245007.png" />. The conditional expectation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245008.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c0245009.png" /> is understood to be a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450010.png" />, measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450011.png" /> and such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450013.png" />. If the expectation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450014.png" /> is infinite (but defined), i.e. only one of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450016.png" /> is finite, then the definition of the conditional expectation by means of (*) still makes sense but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450017.png" /> may assume infinite values.

The conditional expectation is uniquely defined up to equivalence. In contrast to the [[Mathematical expectation|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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450018.png" /> if, almost certainly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450019.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450020.png" /> for every real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450021.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450022.png" /> for arbitrary real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450024.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450025.png" />;

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450026.png" /> for every convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450027.png" />. Furthermore, the following properties specific to the conditional expectation hold:

6) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450028.png" /> is the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450029.png" />-algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450030.png" />;

7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450031.png" />;

8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450032.png" />;

9) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450033.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450035.png" />;

10) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450036.png" /> is measurable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450038.png" />.

There is a theorem on convergence under the integral sign of conditional mathematical expectation: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450039.png" /> is a sequence of random variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450041.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450043.png" /> almost certainly, then, almost certainly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450044.png" />.

The conditional expectation of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450045.png" /> with respect to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450046.png" /> is defined as the conditional expectation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450047.png" /> relative to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450048.png" />-algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450049.png" />.

A particular case of the conditional expectation is the [[Conditional probability|conditional probability]].

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



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Ash,   "Real analysis and probability" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neveu,   "Discrete-parameter martingales" , North-Holland  (1975)  (Translated from French)</TD></TR></table>