1
$\begingroup$

My book gives the following definitions for conditional expectation:

$E(g(X,Y)|Y=y) = \int_{-\infty}^{\infty} g(x,y)f_{X|Y}(x|y) \, dx $

And similarly for the discrete case. But then the exercise questions ask about things like

$E(X|X+Y=z) $

I feel like this hasn't been defined. I only have the definition for the conditional expectation given that $\{{Y=y}\}$, I don't know the definition for the conditional expectation given an arbitrary event (although I can guess what it should be, do I just replace $f_{X|Y}(x|y) $ with the density function of $X$ given the event $\{{X+Y=z}\}$?)

$\endgroup$
1
$\begingroup$

You can introduce a new random variable $Z = X + Y$. Now your problem is to evaluate $ \mathbb{E}[X|Z = z] $, which should have been defined.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.