The moment generating function (MGF) of a geometric distribution with parameter p is given by M(t) = (1 – p) / (1 – pe^t). It is used to find moments, such as the mean (μ = 1/p) and variance (σ^2 = 1/p^2), which characterize the distribution. The MGF’s properties allow for easy computation of higher moments and analysis of the distribution’s behavior. This makes it a convenient tool for modeling discrete random variables that represent the number of trials until a desired outcome, such as in quality control or queuing theory.
