Poisson distribution questions

[silentone]

Homework Statement

Suppose x has a Poisson $\lambda$ distribution

Find the probability generating function and range it is well defined. Then evaluate E[x(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-7)(X-8)(x-9)(x-10)(x-11)]

Homework Equations

f_x (x) = exp(-lamda) (lamda)^x/x! for x=0,1,2,3...

The Attempt at a Solution

The probability generating function I got easily by using the exponential power series and got p_x (s) = exp(lamda(s-1)) . It is well defined for s real.

I do not know how to approach the expected value.

 

[phiiota]

I could be wrong on this, so take this with a grain of salt.

Apply the definition of expected value. It looks like you could rewrite your term as x!/(x-12)! Does this help? Then when you take the sum, the x! should cancel, and you'll be left with (x-12)! on the bottom.