# Math Help - E[ | X - Y | ] where X and Y are independent Poisson random variables

1. ## E[ | X - Y | ] where X and Y are independent Poisson random variables

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.

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.

2. ## Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

what are the means of these Poisson's?

3. ## Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

Was going to write the answer in terms of E[X] and E[Y] but if not could use:

mean of X : E[X] = \lambda^X dt
mean of Y : E[Y] = \lambda^Y dt

4. ## Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

I dont follow
X is a random variable, the means are random?
And the dt is?

5. ## Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

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.