1014 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



as in Section 1.4; a pair (x, z) of pattern points contributes one coinci- 

 dence to the total number A^ only if both | a; — 2 | < 5 and the interval 

 between x and z is empty. 



Note that Fo{L) is related to the distribution function for the mini- 

 mum distance between the points of the pattern in (0, L) : 



Prob (min. dist. ^6) = 1 -Foil), 



where it must be remembered that Fo(L) is a function of 8. 



As in Part I, we first define the conditional probabilities Pn{L) = 

 Prob (exactly A^ coincidences in (0, L), given a point at L). We then 

 have the following equations: 



UL a, PAL) = 



If L > 6, and A'' ^ 1, the probability of exactly N coincidences in (0, L) 

 equals the probability of A^ coincidences up to the last point of the pat- 

 tern in the interval (0, L) — and if there are to be any coincidences, there 

 must be points of the pattern in (0, L). Hence, if L > 5, A^ ^ 1, 



Fn{L) = \' PAL - y)e-^'X dy. (2-3) 



If A^ = 0, the same argument applies, but there is also the possibility 

 that there are no points at all of the pattern in (0, L). Hence, if L > 5, 



F,{L) = e-^' + f ' Po(L - y)e-^\ dy. (2-4) : 



Now let us consider the case where there is a point of the pattern at L. 

 Then if the last point preceding L is between L — 8 and L, this point 

 and the point at L will create a coincidence; if there is no point within 

 (L — 8, L), then all coincidences are within (0, L — 8). Hence, if L > 5, 

 and A^ ^ 1, 



p^L) = [ Ps-x{L - y)\e-'"dy + f-'V^(L - 8). (2-5) 

 Jo 



For the case A^ = 0, we cannot allow a point in the interval (L — 8, L), 

 and hence, if L > 8, 



P,(L) = e-^'F,{L - 8). (2-6) 



