Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=\lambda^2\exp(-\lambda y)$.

Question

Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=\lambda^2e^{-\lambda y}$ for $0\leq x \leq y, \lambda >0 $.

I used the Density transformation theorem to find the joint density of $g(s,t)= \lambda^2 e^{-\lambda t}$.

Then I find the marginal density $g(s)= \int g(s,t) dt$ where $s= x + y$, and $t = y$.

However, I struggle with two things:

1. How to determine the range of $g(s,t)$ function?
2. How to determine the lower and upper boundry pf the integral?

Answer 1

The pdf of $Z:=X+Y$ is $$\int_{0\le x\le y}\lambda^2 e^{-\lambda y}\delta (x+y-z)dxdy=\int_{z/2}^z\lambda^2 e^{-\lambda y}dy=\lambda (e^{-\lambda z/2}-e^{-\lambda z})$$for $z=0$. (Note this is clearly a pdf.) I'll explain the steps below.

First we write down an integral using the $\delta$ function; then we integrate out $x$, which only retains value of $y$ for which $x+y-z$ is negative when $x=0$ but positive when $x=y$, so that the integral is around the delta peak. These constraints can be stated as $y\le z\le 2y$, or equivalently $y/2\le z\le y$.