COINCIDENCES IX POISSOX PATTERNS 



1019 



Fig. 5 — Solution of s + X = Xe 



-(s + X)o 



III COINCIDENCES BETWEEN n POISSON PATTERNS 



3.1 Integral Equation 



In this part we consider n one-dimensional Poisson patterns and ask 

 for the probability, F{L), that in the interval (0, L) no pair of points 

 from different patterns are coincident. Unlike Part I, we now consider 

 only the case in which all n patterns have the same density X. Let P{L) 

 be the conditional probability, given that Pattern No. 1 has a point at 

 L, that there are no coincidences in (0, L). 



If ^ L ^ 5, P{L) = exp - (» - l)\L. 



If 5 < L, 



by the same sort of argument used in Part I. Then F{L) will be given 

 bv 



P{L) = e 



-(n-DXL 



( 



1 -\- in - l)\e 



-u 



I 



L-S 



F{L) = g-"'"' (l + nX I"" e"'"P(.T) dx\ 



