1008 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



lowing related papers. C. Domb' finds the distribution function for the 

 total length of the set of points lying within distance 5 of a pattern point. 

 P. Eggleton and W. O. Kermack' and also L. Silberstein" consider ag- 

 gregates, Avhich are sets of k pattern points all contained in an interval 

 of length b. In the special case k = 2, aggregates are our coincidences. 

 These authors find the expected number of aggregates but not the prob- 

 ability of A^ aggregates. 



I CONCIDENCES BETWEEN TWO PATTERNS 



1.1 Integral Equation 



Consider two Poisson patterns of points on the real Une, the first \nth 

 density X (points per unit length) and the second with density n. We want 

 the probability F{L) that in the segment from to L there is no coinci- 

 dence between a point of pattern No. 1 and a point of Pattern No. 2. 

 F{L) will be formulated in terms of the conditional probabilities 



Pi(L) = Prob (no coincidence, given Pattern No. 1 has point at L), 



P^iL) = Prob (no coincidence, given Pattern No. 2 has point at L). 



If L ^ 5, Pi(L) and P2{L) are the probabilities that patterns No. 2 

 and No. 1 are empty: 



Pi(L) = e-'\ Po(L) = e-'\ if L ^ 5. (1-1) 



If L > 6 and Pattern No. 1 contains a point at L, there are two ways 

 that no coincidences can occur. First, Pattern No. 2 may fail to have any 

 points anywhere in the interval [0, L]. The probability of this event is 

 exp — ijlL. The second possibility is illustrated in Fig. 1 (using circles for 

 points of Pattern No. 1 and crosses for points in Pattern No. 2). Pattern 

 No. 2 has points in (0, L) ; the one closest to L is at ?/ < L — 5. Since 

 the interval {y, L) contains no points of Pattern No. 2, the probability 

 of finding this closest point, y, in an interval, dy, is 



exp l-niL - y)]iJL dy. 



The interval (y, y + b) must be free from points of Pattern No. 1 (prob- 



NO COINCIDENCES EMPTY NO CROSSES 



_j . I . 



1 ■ ^/ — 



■^<r^ -^ ^'^ \ 0^ © 



U u + (f L 



Fig. 1 — Patterns without coincidence. 



