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 $X$ $s$ are:


 * $\dfrac {d^k} {\d s^k} \, \map {\Pi_X} s = \lambda^k e^{- \lambda \paren {1 - s} }$

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:
 * $\dfrac {\d^k} {\d s^k} \paren {e^{\lambda s} } = \lambda^k e^{\lambda s}$

Thus we have:

Hence the result.