What is the expected value of the absolute difference of two independent Poisson variables?
E[ |X - Y| ]
Seems like an easy question but I haven't found an easy solution.
I've split the double sum into the correct regions but not sure what to do with the partial sums remaining.
Sum_0^infinity p(x) Sum_0^infinity |X - Y| p(y)
...since p(x,y) = p(x)p(y)
= Sum_0^infinity p(x) [Sum_0^x (X - Y) p(y) + Sum_x^infinity (Y - X) p(y)]
Should get something like | E[X] - E[Y] | + some variance or covariance term, the latter of which will be 0 since X and Y are independent.