Let $X,Y$ jointly distributed with the following joint density function:
$f(x,y)=\begin{cases} ce^{-(x+y)} & \text{ if } y>x>0 \\ 0 & elsewhere \end{cases}$
Now, it is required to obtain the $E[Y|X=2]$.
My approach
First, I obtained the value of $c$ using the total probability law which turned out to be $c=2$. Then, I obtained the individual density or marginal density of $X$ by integrating the joint density with respect to $y$. The density of $x$ is exponential with the parameter $2$. The conditional density of $y$ given $x=2$, then turned out to be:
$\begin{cases} e^{-(y-2)} & \text{ if } y>2>0 \\ 0 & elsewhere \end{cases}$
The expectation of the above distribution function can be obtained as :
$E(y|x=2)=\int_{2}^{\infty}ye^{-(y-2)}dy$.
After solving the above integration, the conditional expectation is obtained as $3$.
Did I do everything correctly? My manual gives the answer $1$. Also, is there a quick way to solve the given problem since I had to solve a lot of integral quantities.
Thanks.