Coincidences in Poisson Patterns 



By E. N. GILBERT and H. O. POLLAK 



(Manuscript received August 3, 1956) 



A number of practical problems, including questions about reliability of 

 Geiger counters and short-circuits in electric cables, reduce to the mathe- 

 matical problem of coincidences in Poisson patterns. This paper presents 

 the probability of no coincidences as well as asymptotic formulas and simple 

 bounds for that probability under a variety of circumstances. The probability 

 of exactly N coincidences is also found in some cases. 



IXTRODUCTIOX 



A number of practical problems are questions about what we call 

 "coincidences" in Poisson patterns. In c?-dimensional space, a Poisson 

 pattern of density X is a random array of points such that each infinitesi- 

 mal volume element, dV, has probability \dV of containing a pomt, 

 and such that the numbers of points in disjoint regions are independent 

 random variables. Then a volume, V, has probability 



(xy)' 



e 



of containing exactly k points. A coincidence, in our usage of the word, 

 is defined as follows: We imagine a certain fixed distance 8 to be given 

 in advance; two points are then said to be coincident if they lie within 

 distance 8 of one another. 



Examples 



The best-known case of a coincidence problem concerns Geiger coun- 

 ters. 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. 



lOUo 



