1024 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



for S H < 1. This follows immediately from the concavity of 



(1 + xr. 



Suppose now that (4-5) holds for a value H. If we integrate both 

 sides of (4-5) from to x, we obtain 



'"'''' -i-'^' rr^rr .^ ^rr , , .^ ^ (1 + :c)"+^ - 1 



X 



+ Z ,, , ,^, H{H-1) ■■• (H - k + 1) ^ 



,^ (/.• + 1)! ^ ' ^ ' ' - H +\ 



which may be rewritten as 



{H+Vi+l k 



1 + Z n (^ + l)(^) •••(/?- A: + 2) ^ (1 + xr+\ 

 This completes the induction, and the proof of the theorem. 



4-3 Another Lower Bound {Any Number of Patterns) 



Another kind of lower bound can be derived which sometimes will 

 be better than the above bounds when the region V has a large fraction 

 of its volume within 5 of the boundary. For example, V might be a 

 three-dimensional circular cylinder (a cable) with a radius which is com- 

 parable to 5. 



To derive this bound one first finds the expected number, E, of co- 

 incidences in V. An upper bound on E will also suffice. Then it is noted 

 that 1 — £" is a lower bound on the probability of no coincidences. For 

 if Qiv is the probability of finding A'' coincidences, 



00 



E = T. NQ^ ^ E Qiv = 1 - Qo . (4-6) 



iV=l 



44 Thick Cable 



For example, we now give a lower bound which is of interest in con- 

 nection with the problem of a cable mth many wires. 



Theorem: Let a Poisson pattern of points with density X be placed in a 

 cylinder of length L and radius R > 8. The probability of finding no co- 

 incidences in the cylinder is at least as great as 



^5 \ 





1 - XV-^L ( ::^^ - i:;! -f- L 



Proof 



Introduce cyUndrical coordinates r, (p, Z so that the cylinder is de- 

 scribed by 



r < R, < Z < L. 



