Density of $g(Y)=\frac{1}{2}\mathbb{E}[X|Y]$

Question

Let $(X,Y)$ a random variable with density $f(x,y)=cx(y-x)e^{-y}$ for $0 \leq x \leq y <\infty$. Find:

• 1) the value of $c$.

$\rightarrow c=1$

• 2) the density of $X|Y=y$.

$\rightarrow f_{X|Y}(x,y):=\frac{f_{XY}(x,y)}{f_Y(y)}=\frac{fx(y-x)}{y^3}$

• 3) the density of the random variable $g(Y)=\frac{1}{2}\mathbb{E}[X|Y]$.

$\rightarrow g(Y)=\frac{1}{4}y$.

Is it correct? In particular the third point? Thanks in advance.

Answer 1

There is a missing factor of 6 in your solution to part 2); I can only assume that this is a typo, and the $f$ was intended to be a 6. Your solution to the third point is indeed correct.