Derivatives of PGF of Poisson Distribution

Theorem
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.

Then the derivatives of the PGF of $X$ w.r.t. $s$ are:


 * $\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \lambda^k e^{- \lambda \left({1-s}\right)}$

Proof
The Probability Generating Function of Poisson Distribution is:

We have that for a given Poisson distribution, $\lambda$ is constant.

From Higher Derivatives of Exponential Function, we have that:
 * $\dfrac {\mathrm d^k} {\mathrm d s^k} \left({e^{\lambda s}}\right) = \lambda^k e^{\lambda s}$

Thus we have:

Hence the result.