COINCIDENCES IN POISSON PATTERNS 1031 



5.1.2 The Asymptotic Formula 



All approximation to the probability F{x) of no coincidences is given 

 by the asymptotic formula (1-10) which, of course, becomes a better 

 approximation the larger L becomes. If X = 5, /x = 10, and 5 = 0.02, 

 the smallest value, a, such that 



(X - a){n - a) = Xixe--^^^'-"'' 



is a = 1.548. The asymptotic formula for F{L) now becomes 



F{L) ^ 1.013e"'•'''^ 



which is found in Fig. 8 as ?/2 . 



5.1.S Bounds Using the Asymptotic Exponent 



Formulas (1-12) through (1-18) give a scheme for computing both 

 upper and lower bounds for F(L) which have the right behavior for 

 large L, and also agree with the solution at L = 0. They become 



F(L) ^ 1.007e~' •'*'"' - 0.007e"''^ 

 and 



FiL) ^ 1.195e~' •'''"' - 0.195^''^ 

 respectively, and are represented by ^3 and yi in Figure 8. 



5.1.4 An Upper Bound by a Discrete Markov Process 



If we mark on the positive a;-axis the points nb/2, ?i = 0, 1, 2, • • •, 

 we can assign to each interval of length 6/2 thus created a state (z^), i, 

 j = or 1, as follows: t = if no point of the X-process is present in the 

 interval, i — 1 if one or more points of the X-process are present, and 

 similarly for j and /x. An interval of length 5, made up of two adjacent 

 intervals of length 5/2, may then be represented by a number between 

 and 15 in binary notation, where 3, 6, 7, 9, and 11-15 represent a 

 coincidence within the interval of length 5. We now define a Markov 

 process as follows: in the interval ^ / < 5, let p/°\ z = 0, 1, 2, 4, 5, 

 8, 10, be the probabilities of occurrence of the i state, so that, for exam- 

 pie, p^ = e-'''e-''\ and p/°' = e-'''e-'\\ - e-''). These are the 

 states in which there is no coincidence in (0, 5). In addition, let g^°^ 

 represent the probability of all the other states put together; i.e., of a 

 coincidence in (0, 5). We now define pi'"\ i = 0, 1, 2, 4, 5, 8, 10 as the 

 probability of the i state in the interval {nb/2, (n + 2)5/2), where we 



