Probability Generating Function of Poisson Distribution

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Theorem

Then the p.g.f. of $X$ is:

$\Pi_X \left({s}\right) = e^{-\lambda \left({1-s}\right)}$

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{x \ge 0} p_X \left({x}\right) s^x$

From the definition of the Poisson distribution:

$\displaystyle \forall k \in \N, k \ge 0: p_X \left({k}\right) = \frac {e^{-\lambda} \lambda^k} {k!}$

So:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \Pi_X \left({s}\right)$	$=$	$\displaystyle \sum_{k \ge 0} \frac {e^{-\lambda} \lambda^k} {k!} s^k$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e^{-\lambda} \sum_{k \ge 0} \frac {\left({\lambda s}\right)^k} {k!}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e^{-\lambda} e^{\lambda s}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by Taylor Series Expansion for Exponential Function
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e^{-\lambda + \lambda s}$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Hence the result.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Pi_X \left({s}\right)\)	\(=\)	\(\displaystyle \sum_{k \ge 0} \frac {e^{-\lambda} \lambda^k} {k!} s^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e^{-\lambda} \sum_{k \ge 0} \frac {\left({\lambda s}\right)^k} {k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e^{-\lambda} e^{\lambda s}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by Taylor Series Expansion for Exponential Function
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e^{-\lambda + \lambda s}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)