Let X and Y be random variables with $\displaystyle p(x,y)=e^ {-y} $ for $\displaystyle 0<x<y<+ \infty $

Find $\displaystyle E[X] $ and $\displaystyle E[E[X|Y]] $

My solution:

a) First of all, $\displaystyle p(x) = e^{-x} $, so I have $\displaystyle E[X] = \int ^ y _0 xe^{-x}dx = -ye^{-y}-e^{-y}+1 $

b) Here, $\displaystyle E[X|Y] = \frac {Y}{2} $ and $\displaystyle p(y)=ye^{-y} $, so I have $\displaystyle E[E[X|Y]] = \int E[X|Y=y]p(y)dy = \int ^ \infty _x \frac {y}{2}(ye^{-y})dy= \frac {1}{2} e^{-x}(x^2+2x+2) $

But according to the law of total expectation, they should be the same thing, so what am I doing wrong? Thank you!