PGF of a Poisson Process

Question

I'm trying to find the probability generating function of a general Poisson Process and am a little stuck.

The PGF is defined as $E(s^{N_t})$, and I know that the density function of $S_n$ is:$$f_{S_n}(t)=\frac{\lambda^n t^{n-1} e^{-\lambda t}}{(n-1)!}, \text{for } t>=0$$

Further, I know PGFs are of the form $E(s^{N})=\sum_{k=0}^\infty S^{k}P(X=k)$

Not sure how to proceed from here though... help appreciated!

Answer 1

Just work it out directly. $P(N_t=k)=e^{-\lambda t}\frac{(\lambda t)^{k}}{k!}$, so:

$$\sum_{k=0}^\infty s^kP(N_t=k)=e^{-\lambda t}\sum_{k=0}^\infty \frac{(\lambda s t)^k}{k!}=e^{-\lambda t}e^{\lambda ts}=e^{\lambda t(s-1)}$$