The index of dispersion of a probability distribution is defined to be the variance-to-mean ratio of the distribution. In this paper we formulate initial value problems involving second order nonlinear differential equations, the solutions of which are moment generating functions or cumulant generating functions of random variables having specified indices of dispersion. By solving these initial value problems we will derive relations between moment and cumulant generating functions of probability distributions and the indices of dispersion of these distributions. These relations are useful in constructing probability distributions having a given index of dispersion. We use these relations to construct several probability distributions having a unit index of dispersion. In particular, we demonstrate that the Poisson distribution arises very naturally as a solution to a differential equation.
Nassor S. Nassor
"Constructing Probability Distributions Having a Unit Index of Dispersion,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 17:
1, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol17/iss1/3