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''conditional expectation, of a random variable''
 
''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
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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
  
<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>
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\begin{equation}\tag{*}
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\int\limits_BX\mathsf{P}(d\,\omega)=\int\limits_B\mathsf{E}(X\, |\, \mathfrak{B})\mathsf{P}(d\,\omega)
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\end{equation}
  
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.
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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|mathematical expectation]], which is a number, the conditional expectation represents a function (a random variable).
 
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).
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The properties of the conditional expectation are similar to those of the expectation:
 
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" />;
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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) <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" />;
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2) $\mathsf{E}(c\, |\, \mathfrak{B})=c$ for every real $c$;
  
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" />;
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450025.png" />;
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4) $|\mathsf{E}(X\, |\, \mathfrak{B})|\leq\mathsf{E}(|X|\, |\, \mathfrak{B})$;
  
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:
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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 <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" />;
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6) If $\mathfrak{B}=\{\emptyset, \Omega\}$ is the trivial $\sigma$-algebra, then $\mathsf{E}(X\, |\, \mathfrak{B})=\mathsf{E}X$;
  
7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450031.png" />;
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7) $\mathsf{E}(X\, |\, \mathcal{A})=X$;
  
8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024500/c02450032.png" />;
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8) $\mathsf{E}(\mathsf{E}(X\, |\, \mathfrak{B}))=\mathsf{E}X$;
  
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" />;
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9) if $X$ is independent of $\mathfrak{B}$, then $\mathsf{E}(X\, |\, \mathfrak{B})=\mathsf{E}X$;
  
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" />.
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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 <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" />.
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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 <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" />.
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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|conditional probability]].
 
A particular case of the conditional expectation is the [[Conditional probability|conditional probability]].

Latest revision as of 11:43, 14 June 2017


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)
How to Cite This Entry:
Conditional mathematical expectation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_mathematical_expectation&oldid=41628
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article