Limits of integration for expectation of absolute value of difference between two exponential variables.

Question

Let $X,Y$ iid and $\sim Exp(\lambda)$.

Find $E[|X-Y|]$

Would someone tell me what the limits of integration are?

Answer 1

Hint: use symmetry.

$$\begin{align}\mathsf E(\lvert X-Y\rvert) &= \mathsf E((X-Y)\mathbf 1_{Y<X})+\mathsf E((Y-X)\mathbf 1_{X\leqslant Y}) \\[2ex] &= 2~\mathsf E((X-Y)\mathbf 1_{Y<X}) \\[2ex] &{= 2~\iint_{y<x} (x-y)\,f_{X,Y}(x,y)~\mathrm d (x,y)} \end{align}$$