Use the method of distribution functions to find the probability density function of $U$

Question

I have been given that: $f(y_1, y_2)= e^{-y_1},0 \leq y_2 \leq y_1 \leq \infty$ and $0$ otherwise.

$U:= y_1 - y_2$

This is my work so far:

$0\leq u$

$F_U(u) = P(U \leq u) = P(y_1 - y_2 \leq u) = P(y_1 \leq u+y_2)$

This is where I become unsure of my work:

$\int_0^\infty \int_0^{u+y_2}[e^{-y_1}]dy_1dy_2 = \int_0^\infty[e^{-y_1}]_0^{u+y_2}dy_2 = \int_0^\infty[-e^{-u-y_2} +1]dy_2 = [e^{-u-y_2}+y_2]_0^\infty$

Seeing as this is an indeterminate form, I assume I have made a mistake with my bounds. Any help would be appreciated.

Answer 1

Your mistake was in forgetting the condition $y_1 \geq y_2$. The inside integral is over the interval $(y_2 , u+y_2)$, not $(0, u+y_2)$. You will now get the right answer.