Probability Generating Function of Poisson Distribution
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Theorem
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.
Then the p.g.f. of $X$ is:
- $\map {\Pi_X} s = e^{-\lambda \paren {1 - s} }$
Proof
From the definition of p.g.f:
- $\ds \map {\Pi_X} s = \sum_{x \mathop \ge 0} \map {p_X} x s^x$
From the definition of the Poisson distribution:
- $\ds \forall k \in \N, k \ge 0: \map {p_X} k = \frac {e^{-\lambda} \lambda^k} {k!}$
So:
\(\ds \map {\Pi_X} s\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} \frac {e^{-\lambda} \lambda^k} {k!} s^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^{-\lambda} \sum_{k \mathop \ge 0} \frac {\paren {\lambda s}^k} {k!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds e^{-\lambda} e^{\lambda s}\) | Taylor Series Expansion for Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{-\lambda + \lambda s}\) |
Hence the result.
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.2$: Integer-valued random variables: $(12)$