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BELL SYSTEM TECHNICAL JOURNAL 



the integral of the product of the function by the frequency of occur- 

 rence for each value of it, divided by the integrated frequency of oc- 

 currence alone. The frequency of occurrence of individual values of 

 the function is the same as that of the corresponding values of its 

 argument, and hence can be written as F(h)dh where F{h) is the distri- 

 bution function of the variable h. The average value of Hm is therefore 



Hn, 



HI 



HmFiF2 • • • FndhidJh • • • dhn 



= I I • • • I E (^j - hj-+i){hj+„, - hi+m+i) 



X F1F2 ■ ■ ■ Fndhidh2 ' ' -dhn, (16) 

 where Fk is the distribution curve of hu, and 



£ 

 .£ 





Fkdhk = 1, 



kkFidhk = 0. 

 Assuming the /z's all have equal distribution curves: 



hk^Fkdhk = /^^ 

 except that since ho = and A„+i = 0, then 



JiQ^Fodho = 0, 



and 



Likewise 



except that 



X 



X 



X 



}l~ n+\F n+ldh. n+i = 0. 



r hk'Fkdhk = h\ 



»y — 00 



ho'^Fodho = 0, 



■<J0 



¥n+ldFn+ldhn+l = 0. 



(17) 

 (18) 



(19) 



(20) 

 (21) 

 (22) 

 (23) 

 (24) 



Considering the four products hjhj+m, hjhj+m+i, hj+ihj+m and hj+ihj+m+i, 

 it will be seen that they all integrate to zero by virtue of symmetry 



