Probability Generating Function of Shifted Geometric Distribution
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Theorem
Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$.
Then the p.g.f. of $X$ is:
- $\map {\Pi_X} s = \dfrac {p s} {1 - q s}$
where $q = 1 - p$.
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 shifted geometric distribution:
- $\forall k \in \N, k \ge 1: \map {p_X} k = p q^{k - 1}$
So:
\(\ds \map {\Pi_X} s\) | \(=\) | \(\ds \sum_{k \mathop \ge 1} p q^{k - 1} s^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p s \sum_{k \mathop \ge 1} \paren {q s}^{k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p s \sum_{j \mathop \ge 0} \paren {q s}^j\) | change of index | |||||||||||
\(\ds \) | \(=\) | \(\ds p s \frac 1 {1 - q s}\) | Sum of Infinite Geometric Sequence |
Hence the result.
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.2$: Integer-valued random variables: Example $9$