COINCIDENCES IN POISSON PATTERNS 1013 



For example, in Fig. 3, we would count iV = 6 coincidences even 

 ! though there are 18 pairs which satisfy (i). In cable problems it appears 

 I reasonable to count coincidences as above. If we assume that all flaws 

 are equally bad, then a short circuit is likely to develop only across an 

 , adjacent coincidence; our N is the number of places on the cable at 

 which a short circuit can form. Another interpretation is that the cable 

 can be cut into exactly iV + 1 pieces each of which contain no coinci- 

 dences. 



Let Pi,n{L) be the conditional probability of having N coincidences 

 in (0, L) knowing that there is a point of Pattern No. 1 at L. The Lap- 

 lace transform of Pi,n{L) turns out to be the coefficient of t^ in a generat- 

 ing function of the form 



'& 



VAt, s) - 



(X + s)(m + s) - X/xfi^' 

 where = g-(^+''+«^^Q _/)_!_/ Interchanging X and /x one gets the gen- 

 erating function p2{t, s) for the Laplace transform of the probability 

 P2,n{L) of A^ coincidences, given a point of Pattern No. 2 at L. Finally 

 the Laplace transform of F.w{L) is the coefficient of t^. in the generating 

 function 



1 - 6-^^+^+^^^ + \p,(t, s) + mit, s) 



fit, s) 



X + /i + s 



Since /(/, s) is a rational function of /, it is easy to find the coefficient of 

 t' . The poles of this function are again just zeros of D(s). Now, however, 

 the poles are higher order poles. For large L an asymptotic formula for 

 Fn{L) has the form exp — aL times a polynomial in L with degree de- 

 pending on N. 



For more details about this method we refer the reader to Part II 

 where a similar, but less involved, calculation is carefully done. 



II SELF-COINCIDENCES IN ONE POISSON PATTERN 



2.1 Integral Equation 



In this part we shall consider a single one-dimensional Poisson pattern 

 with density X and ask for the probability Fn{L) that in the interval 

 (0, L) the pattern have exactly iV coincidences. We count comcidences 



I ^^ ^ — I (T:0 X )( )( *e@ 1 



(T L 



Fig. 3 — Patterns with six coincidences. 



