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.

I have:

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.

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.

I have:

$$\sum_0^\infty p(x) \sum_0^\infty |X - Y| p(y)$$

...since p(x,y) = p(x)p(y)

$$= \sum_0^\infty p(x) [\sum_0^x (X - Y) p(y) + \sum_x^\infty (Y - X) p(y)]$$

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.