# Find some expectation of two random variable given the joint distribution

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

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:

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.

• 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. – whuber Nov 12 '20 at 16:27
• Yes, sorry, it's my fault, I've edited the question – docdev Nov 12 '20 at 16:38

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.