3
$\begingroup$

I have the following joint distribution for the random variable (X,Y):

enter image description here

Now, I have to compute the following expectations:

  1. $E(X|Y)$
  2. $E(X+Y^2 | Y)$

Now, for the first point, the procedure was the following:

enter image description here

But I don't know how to do the second one. Intuitively maybe I can write something like:

$E(X+Y^2 | Y) = E(X|Y) + E(Y^2 | Y)$

but I stopped here.

Edit 1: I forgot to specify that $x,y$ are positive.

Improve this question
$\endgroup$
  • 2
    $\begingroup$ This is not a valid joint distribution function. You need to be clear about its domain of definition. Presumably $y$ and $x$ must be positive -- but that's not the only possible choice. $\endgroup$ – whuber♦ Nov 12 '20 at 16:27
  • $\begingroup$ Yes, sorry, it's my fault, I've edited the question $\endgroup$ – docdev Nov 12 '20 at 16:38
3
$\begingroup$

Assuming the distribution is valid, for which you need to clarify its support, you've already solved it: $$\mathbb E[X+Y^2|Y]=\mathbb E[X|Y]+\mathbb E[Y^2|Y]=1+1/Y+Y^2$$ Because given $Y$, expected value of $Y^2$ is itself.

Improve this answer
$\endgroup$

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.