Chains of Rare Events

Abstract

The negative binomial distribution of Greenwood and Yule is generalized and modified in order to obtain distribution curves which could be used in many concrete cases of chains of rare events. Assuming that the numbers of single, double, triple, and so on, events are distributed according to Poisson's law with parameters $\lambda_1, \lambda_2, \lambda_3 \cdots$ respectively, and that $\lambda_s$ is given by $\lambda_s = \lambda_1 \frac{a^{s - 1}}{s!}$, the probability of obtaining $M$ successful events is studied. In the considered relation $\lambda_s$, for convenient values of $a$, first increases with $s$ and after a certain saturation value of $s$ starts to decrease. A relation of this type is very suitable for studying the distribution of score in a match between two first class billiard players, the probability of accidents on a highway of dense traffic, etc. The general methods of finding the distribution curves for arbitrary relations between the $\lambda$'s are indicated. The method of steepest descent is applied to find an acceptable approximation of the distribution function; and the advantage of this method is pointed out for other similar cases, in addition to the concrete one which was developed, in which the method of direct expansion into power series becomes inapplicable.