Discrete probability distributions and their recursive relations

Abstract

The objective of this project is to study discrete probability distributions and their recursive patterns. In Chapter I, we state the importance of expressing probability distributions in terms of recursive relations. This is because in some certain probability distributions it is often easier to deal with the recursive relations rather than the distributions themselves in obtaining the moments. In Chapter II, we reviewed the various methods for determining discrete probability distributions. Some of these methods include both the binomial and exponential expansions, the Jacobian transformation (Change of Variable technique) .We also applied the expectation and convolution approaches to sums of iid random variables to obtain the resulting compound distributions. In Chapter III, we have derived the recursive ratios and recursive relations for the various probability distributions that have been identified in Chapter II. Using the recursive relations, we have obtained the means and variances, based on the pgftechnique (where possible) and Feller's method. In Chapter IV, we reviewed a number of patterns of recursive relations; the main ones being the Panj er (1981) and Willmot (1988) patterns. With these patterns, we have been able to identify the corresponding probability distributions. In Chapter V, we have reviewed-some maximum likelihood estimation procedures. We have applied Sprott's procedure for deriving maximum likelihood equations. Chapter VI contains the conclusions of this project and recommendations for further research.