Two aspects of the Poisson process


Thomas M. Semkow,, Wadsworth Center, New York State Department of Health and University at Albany (SUNY), P.O. Box 509, Albany, NY 12201-0509
The radioactive decay chain equations were solved for a case when all the decay constants λ are equal. We found that the average number of atoms of the chain members is governed by the Poisson distribution. The physical origin of this result lies in the renewal process. An atom that has decayed in the chain with equal decay constants is renewed, that is, it continues to decay with the same decay constant, only its position index in the chain has incremented. The mean of this Poisson distribution is λt, which can take any value.

The distribution of the Poisson parameter is given by the gamma distribution upon a single or multiple sampling, assuming a uniform prior. We present new derivations for multiple sampling, using uniform and scale priors. We obtain gamma distributions shifted with respect to each other. Moments are calculated. Both pictures converge to a maximum likelihood estimator when number of samplings is large. For a single sampling, both pictures result in entirely different predictions for small number of counts.