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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.

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    $\begingroup$ 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
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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.

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