Probability Generating Function of Geometric Distribution
Jump to navigation
Jump to search
Theorem
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$.
Then the p.g.f. of $X$ is:
- $\map {\Pi_X} s = \dfrac q {1 - p s}$
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 geometric distribution:
- $\forall k \in \N, k \ge 0: \map {p_X} k = q p^k$
So:
\(\ds \map {\Pi_X} s\) | \(=\) | \(\ds \sum_{k \mathop \ge 0} q p^k s^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds q \sum_{k \mathop \ge 0} \paren {p s}^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds q \frac 1 {1 - p s}\) | Sum of Infinite Geometric Sequence |
Hence the result.
$\blacksquare$