conditional expectation, covariance

Question. Let X and Y be random variables with joint density

Derive the conditional density, , and the conditional expectation, E[X|Y]. Hence or otherwise, evaluate E(X) and Cov(X,Y).

Answer.

(I highlighted the questions that came up along the answer write up by ->>>>>)

,

.

->>>>>>>Is this support right, or should I say (ie include x in it)

Then expected value

( )

Therefore E(X|Y)=Y-1

"Hence" somehow hint on using E(X|Y) in evaluating E(X), and I try that:

->>>>>>So still need to do the integration to find E(X) directly from the integral, or E(Y) first and then deduct 1 to get E(X). Right?

I find E(Y) since the integration is slightly easier.

but I probably messed up signs somewhere and I feel it should be k... (no big deal, I'll recheck).

->>>>>The question is, do I need to take into account the bound on y which is y<x?

Once I get E(Y), .

Then and again I need to integrate to find E(XY), right? Or there is an easy way...