Coincidences in Poisson Patterns

A number of practical problems are questions about what we call "coincidences" in Poisson patterns. In (/-dimensional space, a Poisson pattern of density X is a random array of points such that each infinitesimal volume element, dV, has probability dV of containing a point, and such that the numbers of points in disjoint regions are independent random variables. Then a volume, V, has probability (XV)k - r r e of containing exactly k points. A coincidence, in our usage of the word, is defined as follows: We imagine a certain fixed distance 5 to be given in advance; two points are then said to be coincident if they lie within distance 5 of one another. Examples The best-known case of a coincidence problem concerns Geiger counters. In the simplest mathematical model, there is a short dead-time 5 after each count during which other particles can pass through the counter without registering a count. In our present terminology, a count is missed whenever two particles traverse the counter with coincident times of arrival. The same problem is encountered with telephone call registers. 1005