Abstract
It is a well known fact that for the hierarchical model of a Poisson random variable Y whose mean has an Erlang distribution, the unconditional distribution of Y is negative binomial. However, the proofs in the literature [1] [2] provide no intuitive understanding as to why this result should be true. It is the purpose of this manuscript to give a new proof of this result which provides such an understanding. The memoryless property of the exponential distribution allows one to conclude that the events in two independent Poisson processes may be regarded as Bernoulli trials, and this fact is used to achieve the research purpose. Another goal of this manuscript is to give another proof of this last fact which does not rely on the memoryless property.
Highlights
There is much current interest in compounding or mixing distributions and their applications
The memoryless property of the exponential distribution allows one to conclude that the events in two independent Poisson processes may be regarded as Bernoulli trials, and this fact is used to achieve the research purpose
Two independent Poisson processes are carefully chosen and the memoryless property of their exponentially distributed interarrival times is used to conclude that the events in these processes may be regarded as Bernoulli trials
Summary
There is much current interest in compounding or mixing distributions and their applications. The work by Greenwood and Yule [4] in the more modern era has been followed up with new results and extensive applications [5], including many based on the Poisson distribution because of its centrality in statistical analysis and probability modeling [6] [7] [8] [9]. An integration of the product of (1) and (2) yields (3) This result appears in many different settings Two independent Poisson processes are carefully chosen and the memoryless property of their exponentially distributed interarrival times is used to conclude that the events in these processes may be regarded as Bernoulli trials.
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