Hi, I think I have most of this question correct, so mostly I just want someone to check my results, but I am stuck on the last part, so will need some help with that, thanks.

1) Find the marginal pdf of X. I worked that out by integrating out y to give

i possibly have the limits wrong here?

2) Determine whether X and Y are independent.

I went about this by finding out if

I worked out which therfore means X, and Y are not independent.

3)Calculate the covariance between X and Y

I got

so

I did the same method to get

To get i did

Which i split to

Which came out as

So

this could all possibly be wrong, although the result is within -1..1 so perhaps i am correct.

4)Find the conditional probability density function of Y given X=x. Hence fin the conditional expectation E(Y|X=x)

This is the one i am stuck on. I know

However i am finding it tough to simplify this, and even if i had a simplified result, i would be stuck on how to work out the expectation. Any help at all will be fab, thanks.