Experimenting with the generalized binomial distribution

Abstract

The standard binomial distribution is based on a sequence of Bernoulli trials. The probability of success is the same for each trial. In the Generalized Binomial Distribution (GBD), the probability of success changes by the history of successes up to that point. For example, if a student correctly answers a large proportion of the first few questions, the probability of correctly answering the next question increases because the history of successes may indicate skill. The Bernoulli events are correlated. Alternatively, the standard binomial distribution does not consider the history of successes, and assumes that successes are not correlated. In this paper we prove some new properties of the GBD and apply the concept to five new areas in sport competition, finance, biology, meteorology, and testing pseudo-random number generators. We prove that the GBD predicts the results in the collected data significantly better than the standard binomial distribution thus proving that the events are indeed correlated. We provide an R code, and a supplementary Excel file that provide users with an easy way to analyze any collected data set based on Bernoulli processes.