COINCIDENCES IN POISSON PATTERNS 1033 



about L = 0.5 (25 iterations), when the integral equation upper bound 

 becomes better. 



5.2 A Single Pattern in a Square 



To test our higher-dimensional bounds, we consider again coincidences 

 in a single Poisson pattern in a square of side 25. The exact probability 

 of no coincidences was given in l^art JV assuming square neighborhoods. 

 The lower bound (Sec. 4.2) 



e-'''{l + \S{8)V"''> 



applies using T^ = (25) and aS(5) = (26)' for square neighborhoods. To 

 use the lower bound 1 — E we note that the exact expected number of 

 coincidences is 



1 , r'-^ r" 



£; = -X' / / A{x, y) dxdij 



where A{x, y) is the area of the intersection of the given sc^uare with 

 the square neighborhood centered at (x, y). The lower bound is 1 — £" = 

 1 — 9X"5 /2. The upper bound p can be used if the square is cut into 

 /v = 4 squares of side 5, each with a probability?? = (1 + X5") exp — 

 X5" of no coincidence. 



These bounds, together with the exact probability, are plotted as 

 functions of X5' in Fig. 9. When X5"' is small, the 1 — E bound is correct 

 to terms of order 0(X'^5 ). This might have been predicted from (4-6) 

 since it seems reasonable that Q2 , Qi , • ■ • should be of higher order in 

 X than Qi when X is small. Ultimately the first lower bound becomes a 

 better estimate. It must be recognized that this other lower bound is 

 being tested under very severe conditions. Since every point of the 

 square has a neighborhood which intersects the boundarj^, the errors 

 from source (b) of Part ^" are considerable. 



The authors wish to thank D. W. Hagelbarger and H. T. O'Neil for 

 their assistance in the course of the calculations reported in this section, 

 and Aliss D. T. Angell for preparing some of the figures. 



REFERENCES 



I.e. Domb, The Problem of Random Intervals on a Line, Proc. Cambridge Phil. 

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2. P. Eggleton and W. O. Kermack, A Problem in the Random Distribution of 



Particles, Proc. Royal Soc. Edinburgh, Sec A, 62, pp. 103-115, 1944. 



3. W. Feller, On Probabilit}' Problems in the Theor}' of Counters, Studies and 



Essays presented to R. Courant, Interscience, pp. 105-115, 1948. 



4. E. C. Xlolina, The Theory of Prol)abilitv and Some Applications to Engineering 



Prol)lems, Trans. A. I.E. E., 44, pp. 294-299, 1925. 



5. L. Silberstein, The Probable Number of Aggregates in Random Distributions 



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