1. ## conditional expectation

It's me again, sorry...

I've a Problem with a conditional expectation:

$\displaystyle X_1,X_2$ are two random variables and $\displaystyle c>0$ is just a constant.

Does the follwoing hold?

$\displaystyle E[X_1X_2|X_1=c]=cE[X_2|X_1=c]$?

Or does it just hold if $\displaystyle X_1$ and $\displaystyle X_2$ are independent?

2. ## Re: conditional expectation

Hello,

In this specific case, E[X1X2|X1=c]=E[cX2|X1=c]=cE[X2|X1=c]

And in general, X and Y not necessarily independent, E[XY|X]=XE[Y|X]

3. ## Re: conditional expectation

Thank you so much. I'm a litte bit counfused right now..

It's just a special case of the well known property of the conditional expectation:
$\displaystyle E[XY|\mathcal{G}]=XE[Y|\mathcal{G}]$ if $\displaystyle X$ is $\displaystyle \mathcal{G}-$measurable and $\displaystyle \mathcal{G}$ is a $\displaystyle \sigma-$field

Right?

4. ## Re: conditional expectation

Exactly !

When one conditions with respect to a random variable : E[.|X], it's actually conditionned with respect to the sigma-field/algebra generated by the random variable :

$\displaystyle E[.|\sigma(X)]:=E[.|X]$

And by definition, X is $\displaystyle \sigma(X)$-measurable

5. ## Re: conditional expectation

If you also have independence, then

$\displaystyle \mathbb E(XY|X)=X\mathbb E(Y|X)=X\mathbb E(Y)$