A Review of the Convolution of Geometric Distributions and Its Properties

Abstract

This review explores the convolution of geometric distributions, a key operation in probability theory for deriving the distribution of the sum of independent random variables. Geometric distributions quantify the number of Bernoulli trials needed for the first success and are foundational in discrete probability models. Convolving multiple geometric distributions with a common success probability produces a negative binomial distribution, modelling the number of trials needed to achieve a given number of successes. We present a concise derivation of this result, highlighting the relationship between geometric and negative binomial distributions. The review also outlines essential properties of the negative binomial distribution, including its mean, variance, moment-generating function, and some applications.