Probabilities for the local behavior of poisson distributed events
Franz Streit · Scandinavian Actuarial Journal · 1972
The probability law of Poisson, which is specified by the formula assigns to every non-negative integer n the probability Φ(n, u). This distribution has many applications; it is successfully used to describe and explain various random phenomena observed in practical experiments. On the other hand, it plays a very important role in the modern theory of probability. In view of these facts it seems worthwhile to investigate closely the random processes related with the Poisson distribution. In this article we study the local behavior of Poisson distributed events. To our knowledge such problems have not yet found widespread attention. It may therefore be advantageous to indicate shortly the usefulness of such investigations: It is well-known that the Poisson distribution is only suitable to represent actually realized random processes within a limited range of time and space [6, 23, 29, p. 79]. The restriction of the scope of the investigation to intervals makes allowance for this fact. Furthermore many statements concerning the frequency of irregularities of a realization of the Poisson process are only sensible and non-trival when the local and temporal range of the reference-set is limited.