Finding Conditional expectance, variance from joint pdf

Question

X and Y have the joint pdf:

$f(x,y)=e^{-y}$ for $0<x<y<\infty$

Compute $E(Y|X)$ and $Var(Y|X)$.

I computed $f_X(x)= \int_{x}^{\infty} f(x,y)dy=e^{-x}$

$f_{Y|X}= e^{x-y}$ for $ x>0$

Then $E(Y|X)=\int_{0}^{\infty} ye^{x-y}dy = e^x$ But the value for E(Y|X) is given as $1+X$.

Also, I think I can compute $Var(Y|X)=E(Y^2|X)-[E(Y|X)]^2$

Any help would be appreciated

Answer 1

\begin{align} f_{Y|X}(x,y)&=\frac{f(x,y)}{f_X(x)}\ \ \ \text{ where }\ \ f_X(x)\ne0\\ &=\cases{e^{x-y}&if $\ 0\le x\color{red}{\le y}\ $\\ 0&otherwise}\\ \therefore\ \ \ E(Y|X)&=\int_\color{red}{X}^\infty ye^{X-y}dy\ , \end{align} and you're formula for the variance is correct, and probably the easiest way of calculating it.