Thread: conditional expectation

It's me again, sorry...

I've a Problem with a conditional expectation:

are two random variables and is just a constant.

Does the follwoing hold?

?

Or does it just hold if and are independent?

Thanks in advance!

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]

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:
if is measurable and is a field

Right?

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 :



And by definition, X is -measurable

If you also have independence, then