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Factorial Moments of the Geometric Distribution of Order $k$

S. R. Mane

Abstract

We derive a simple expression for the $r^{th}$ factorial moment $μ_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in(0,1)$ (and $q=1-p$) in terms of its probability mass function $f_k(n)$. Specifically, $μ_{(r)} = r!f_k((r+1)k+r)/((qp^k)^{r+1})$.

Factorial Moments of the Geometric Distribution of Order $k$

Abstract

We derive a simple expression for the factorial moment of the geometric distribution of order with success parameter (and ) in terms of its probability mass function . Specifically, .
Paper Structure (1 section, 1 theorem, 12 equations)

This paper contains 1 section, 1 theorem, 12 equations.

Table of Contents

  1. Addendum 12/30/2023

Key Result

Proposition 1

The $r^{th}$ factorial moment $\mu_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in(0,1)$ (and $q=1-p$) is given as follows, for $r\ge1$.

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4