74 



values 



BELL SYSTEM TECHNICAL JOURNAL 

 n= Y. avcUi"'- + Bl] = {2X + 1)(1 + r) 



n=-N 



N 



xo = E ave. Bl = (2.V + l)r 



(5-2) 



Here we keep the convention a = 1, v'/a- = r used in Section 4. The same 

 sort of reasoning as used to establish (5-2) shows that the spread about these 

 average values is given by 



ave. (u - u)- = (4iV + 2)(1 + rY 



ave. (xo — a-,))- = (4.V + 2)r- 



ave. (u — u){xo — xo) = (4.V + 2)r- 



(5-3) 



If the parameters A^, K, and r in the integral (4-12) are such that its value 

 is appreciably different from zero, most of the contribution comes from the 

 region around I'l and .vo where />o(w, .Vo) is appreciably different from zero. 

 However, instead of taking m and fo as reference values, we take the nearby 

 values 



u.2= u - 2 - 2r = (2N - 1)(1 -\- r) = 2q(l + r) 



Xo = .vn — 2r = {2A — l)r = 2qr 



as these turn out to be better representatives of the center of the distribu- 

 tion. We have introduced the number 



q = N 



1/2 



(5-5) 



in order to simplify the writing of later equations. We assume ^ > 1. 

 First, we shall show that 



Prob. (PyQ, ■■■ ,F^Q> PoQ) 



= / ' du ' dxopoiu, .Vo)[l - P{xo , li)]" + ^1 



J uo—a Jx'j—b 



(5-6) 



where a = 2(1 + r){2q log </)'/-, b = 2r{2q log </)'/'- and Ri is of order l/q 

 (denoted by 0(1/^)), i.e. a constant C and a value (/n can be found such that 

 I i?i I < C/q when q > qo. From (4-12) it is seen that Ri is positive and less 

 than 



f du -\- du / dxopo(u,Xo) 



J U2+a J •'0 



+ 



' r/.vii + / (/.\„ / di<pu{u, : 



Jx-y^li J ''0 



(5-7) 



r„) 



