COINCIDENCES IN POISSON PATTERNS 1007 



exact results are available and are illustrated in Parts II and III. Un- 

 fortunately, the exact formulas, although they are finite sums, contain a 

 luunber of terms which grows with L. Much of our effort has been di- 

 rected toward finding good, easily computed bounds and asymptotic 

 formulas. 



The probabilities of having exactly iV coincidences are also obtainable 

 but they have more complicated formulas. A detailed derivation is given 

 only in Part II. 



In Part IV, we consider the probability of no coincidence in higher 

 dimensional problems. The methods of Parts I-III fail in higher di- 

 mensions, but we are still able to derive some bounds. An exact formula 

 is derived for the probability of no coincidences within a single two- 

 dimensional Poisson pattern in a rectangle with sides ^ 25. We also give 

 particular attention to coincidences in a three-dimensional cylinder. 



Part V contains numerical results. 



Reduction of the Examples to the Theory 



We now wish to see how answers bearing on the practical problems 

 previously listed may be found from this stud3^ 



The literature on Geiger counters (see bibliograph}^ in Feller ) is con- 

 cerned with statistics of the number of counts registered in a given long 

 time, /. The basic problem is to test the hypothesis that the particles 

 arrive in a Poisson sequence. To this problem, then, are relevant the 

 formulas for the probability of N coincidences in one pattern given in 

 Part II, and the bounds and asymptotic results there derived. 



The problem of coincident flaws in an electric cable is three-dimen- 

 sional, and we have various approaches leading to the probability of no 

 coincidences which are valid under different circumstances. If the cable 

 contains only two wires (with possibly different flaw densities), then the 

 problem reduces to the one-dimensional case of coincidences between 

 two Poisson patterns treated in Part I. If the diameter of the cable is 

 small with respect to 8, and if the density of flaws is the same on each of 

 the n wires in the cable, we have the situation of )i identical patterns 

 treated in Part III. If, in addition, n is very large, we may ignore the 

 fact that coincident flaws on a single wire do not cause short circuits, 

 and think of coincidences within a single pattern (Part II). AVithout the 

 assumption that the diameter of the cable is small with respect to 8, the 

 problem is no longer reducible to a one-dimensional form. Section 4.4 

 is especially devoted to thick cable, and to producing a lower bound for 

 the probabilit}^ of no coincidences in this three-dimensional situation. 



The literature on Poisson patterns in a line segment contains the fol- 



