Probability Generating Function of Shifted Geometric Distribution

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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 $q = 1 - p$.


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:

$\forall k \in \N, k \ge 1: \map {p_X} k = p q^{k - 1}$


\(\ds \map {\Pi_X} s\) \(=\) \(\ds \sum_{k \mathop \ge 1} p q^{k - 1} s^k\)
\(\ds \) \(=\) \(\ds p s \sum_{k \mathop \ge 1} \paren {q s}^{k - 1}\)
\(\ds \) \(=\) \(\ds p s \sum_{j \mathop \ge 0} \paren {q s}^j\) change of index
\(\ds \) \(=\) \(\ds p s \frac 1 {1 - q s}\) Sum of Infinite Geometric Sequence

Hence the result.