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:


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

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:
 * $$\frac {d^k}{ds^k} \left({e^{\lambda s}}\right) = \lambda^k e^{\lambda s}$$

Thus we have:

$$ $$ $$

Hence the result.