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More on Random Variables

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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

A more general definition of conditional expectation

We will explore conditional expectation in terms of probability spaces. Suppos $\Gc$ is a sub $\sigma$ -field of $\Fc$ . A function $g: \Omega \rightarrow \Rbb$ is ''measurable with respect to $\Gc$ '' if

\begin{displaymath}\{\omega \in \Omega: g(\omega) \in B\} \in \Gc \text{ for all } B
\in \Bc

(Any such function $g()$ would be a r.v. too. But this $g()$ is more restricted.)

We will now define ''conditioning on a $\sigma$ -field.'' Suppose $Y$ is a r.v. with $E[\vert Y\vert^2] < \infty$ and $\Gc$ is a sub $\sigma$ -field of $\Fc$ . Then $E[Y\vert\Gc]$ is any $\Gc$ -measurable random variable such that

\begin{displaymath}\int_A E[Y\vert\Gc] dP = \int_A Y dP \text{ for all } A \in \Gc.

If $Y$ itself were $\Gc$ measurable it would be its own conditional expectation.
Let $(\omega,\Fc) = (\Rbb,\Bc)$. Let $\Gc =
...c{\int_{-\infty,0)} Y dP}{\int_{(-\infty,))} dP}

If $X$ is an r.v., define $\sigma(X)$ (the $\sigma$-field ...
... X(\omega) \in B\} \text{ for } B \in \Bc
Fact: A r.v. $Y$ is measurable with respect to $\sigma(X)$ if and only if there is a measurable function $g:\Rbb \rightarrow
\Rbb$ such that $Y = g(X)$ .

We now define conditional expectation with respect to a $\sigma$ -field:
If $X$ and $Y$ are r.v.s with $E[\vert Y\vert] < \infty$,
...n{displaymath}E[Y\vert X] = E[Y\vert\sigma(X)]

  1. By the fact stated above, we can write

    \begin{displaymath}E[Y\vert X] = g(x)

    for some function $g$ , $g(x) = E[Y\vert X=x]$ .
  2. $E[Y] = E[E[Y\vert X]]$
  3. If $Y$ itself is $G$ -measurable, then $E[Y\vert\Gc] = Y$ .
  4. $E[\alpha Y_1 + \beta Y_2\vert\Gc] = \alpha E[Y_1\vert\Gc] + \beta
E[Y_2\vert\Gc]$ .
  5. If $Y \geq 0$ , then $E[Y\vert\Gc] \geq 0.$
  6. If $E[\vert Y\vert] < \infty$ and $\Gc \subset \Ec \subset \Fc$ , then

    \begin{displaymath}E[E[Y\vert\Ec]] = E[Y\vert\Gc].

    Idea: If you first condition on a field that is less ''course'' than $\Gc$ you get a r.v. Then condition on $\Gc$ .

Two $\sigma$-fields $\Gc$ and $\Hc$ are {\bf independent} ...
...xt{ for all } G \in \Gc \text{ and }H \in \Hc.
Note: $X$ and $Y$ are independent r.v.s iff $\sigma(X)$ and $\sigma(Y)$ are independent $\sigma$ -fields.

  1. If $\sigma(Y)$ is independent of $\Gc$ then $E[Y\vert\Gc] = E[Y]$ .

  2. If $Y$ is $\Gc$ -measurable then $E[Y\vert\Gc] = E[Y]$ .

    So, for example, if $\Gc = \sigma(X)$ , and $Y = g(X)$ for some $g:\Rbb \rightarrow
\Rbb$ (that is, $Y$ is $\Gc$ -measurable) then

    \begin{displaymath}E[Y\vert X] = Y = g(X).\end{displaymath}

    More informally, $E[g(x)\vert X] = g(x)$ .
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). More on Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: This work is licensed under a Creative Commons License Creative Commons License