Suppose $$ X_i, Y_j \sim Exp(\lambda)$$.
$X_i, Y_j$ iid $\forall i,j$
By Symmetry
$$\begin{align}\mathsf E(\lvert X-Y\rvert) &= \mathsf E((X-Y)\mathbf 1_{Y<X})+\mathsf E((Y-X)\mathbf 1_{X\leqslant Y}) \\[2ex] &= 2~\mathsf E((X-Y)\mathbf 1_{Y<X}) \\[2ex] &{= 2~\iint_{y<x} (x-y)\,f_{X,Y}(x,y)~\mathrm d (x,y)} \end{align}$$
I want to find:
$$E(|\sum X_i - \sum Y_j|)$$.
May this be solved by the application of the symmetry formula?
Therefore $$E(|\sum X_i - \sum Y_j|)=E((\sum X_i -\sum Y_j) \mathbb{1}_{Y<X}+E((\sum Y_j -\sum X_i) \mathbb{1}_{X\leq Y}=...?$$
Assuming the expectation of the sum or random variables is the sum of the expectations for iid.
Hence, $$E(|\sum X_i - \sum Y_j|)=\sum E((X_i - Y_j) \mathbb{1}_{Y < X}) + \sum E((Y_j - X_i) \mathbb{1}_{X \leq Y})?$$
Does the triangle inequality have any use here or is it only useful for bounding and approximation? ?
Does minkowski's inequality have any use here?