The p.g.f of a random variable is:

$\displaystyle G_T(x) = exp^{n \theta(x-1)} $

This seems to simplify to:

$\displaystyle f(x|\theta) = \frac{exp^{-n \theta}.\theta^t}{\prod x_i!}$ where $\displaystyle t = \sum x_i $

But this means $\displaystyle exp^{n \theta t} = \frac{\theta^t}{ \prod x_i!} $,

which doesn't look right.

Not sure how the $\displaystyle f(x|\theta)$ equation is derived.