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.

Author Bio

Omar Talib will be graduating with a BSc in statistics next year. After his graduation, he shall apply for a graduate program in pure mathematics and is hoping to get a PhD in some area of pure mathematics. He spends most of his free time studying mathematics and philosophy. He also enjoys very much having long walks and listening to classical music, especially the music of Bach and Beethoven.