What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Question

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.

Answer 1

A less elegant but equally valid approach is to note that we may equivalently write $$Z = |X - Y|,$$ so that $$\begin{align*} f_Z(z) &= \int_{y=0}^\infty f_X(y+z)f_Y(y) \, dy + \int_{y=z}^\infty f_X(y-z)f_Y(y) \, dy \\ &= \int_{y=0}^\infty \lambda e^{-\lambda(y+z)} \lambda e^{-\lambda y} \, dy + \int_{y=z}^\infty \lambda e^{-\lambda(y-z)} \lambda e^{-\lambda y} \, dy \\ &= \lambda e^{-\lambda z}.\end{align*}$$

Answer 2

By symmetry we can assume that $X\lt Y$, so $Z=Y-X$. Now from the memorylessness of the exponential, we have that given that $Y\gt X$, the distribution of $Y-X$ is exponential with parameter $\lambda$.