If joint distribution of $X, Y$ is $e^{-y}I_{0<x<y<\infty}$, find the mgf of $Y-X$

Question

Let joint distribution of $X, Y$ be $e^{-y}I_{0<x<y<\infty}$. Find the mgf of $Y-X$.

Mgf of $x$ is $1/(1-t_1)$ and mgf of $y$ is $1/(1-t_2)^2$.

I know that $x$ and $y$ are not independent, so I decided to use cdf method to calculate the pdf of $y-x$. But I cannot go further. Can anyone help me with this problem?

Answer 1

Notice that the joint density can be written as

$$f_{X,Y}(x,y)=\underbrace{e^{-(y-x)}\mathbf1_{y>x}}_{f_{Y\mid X}(y\mid x)}\cdot \underbrace{e^{-x}\mathbf1_{x>0}}_{f_X(x)}$$

This means $Y-x$ given $X=x$ has a standard exponential distribution. As this conditional distribution is free of $x$, the (unconditional) distribution of $Y-X$ is also standard exponential.

In other words, $Y-X$ and $X$ are independent and identically distributed.