what are the means of these Poisson's?
What is the expected value of the absolute difference of two independent Poisson variables?
E[ |X - Y| ]
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.
No the means are not random, apologies for confusion caused.
Ok, lets just say the means are:
mean X = E[X] = m
mean Y = E[Y] = n
I think the problem can be done in general for any Random variables X and Y.