640 BELL SYSTEM TECHNICAL JOURNAL 



of evaluating the integral, the process was somewhat shortened. In 

 this way equivalent volume distributions have been obtained for 

 w = 2, 4, 8, 16 . . . ; needed points on the distribution curves for 

 intermediate values of n are obtainable by interpolation. 



The accuracy of such a process depends upon the number of division 

 points used in the numerical integration and this as a practical matter 

 must be kept fairly small. When the process must be repeated many 

 times, the errors introduced at each step may accumulate and lead to 

 inaccuracies for large n. It is thus desirable to have some control over 

 the accuracy other than by repeating the calculation with a larger 

 number of division points. This is provided by calculating the 

 moments of pn{Y) from those of pi{W) without the use of numerical 

 integration. 



The moments Sk of pi{W) are defined by 



Su = r Wp.iW) dW, 



and the moments 7"^^"^ of pniY) similarly. By the use of the semi- 

 invariants of Thiele,^ it may be shown that 



Ti^") = nSy, 



Ti^n) = nS2 + n(n - l)Si\ etc. 



By comparing the moments of the distributions obtained by numerical 

 integration with those calculated in this way, and making occasional 

 minor alterations in the curves to bring the first and second moments 

 into agreement, assurance was obtained that all the distributions used 

 are reasonably accurate, with no accumulation of error as n becomes 

 large. 



Examples of the cumulative distribution curves of equivalent volume 

 for 1,4, 16 and 64 active channels are given in Fig. 8. The decrease in 

 standard deviation which occurs as n increases is of interest for it 

 indicates how the fluctuations in load due to talker volume varia- 

 tions are reduced by combining a large number of channels in one 

 system. 



Having now w-channel equivalent volume curves for a range of 

 values of n, the resultant equivalent volume curves may be calculated 

 when the restriction to a fixed number n of active channels is removed. 

 Let pn(V) denote the probability that, with n channels active, the 

 equivalent volume lies between V and V + <iF and let p{n) denote the 

 probability that just n channels will be active. Then the total proba- 



'^ T. N. Thiele, "The Theory of Observations," 1903; reprinted in Annals of 

 Mathematical Statistics, Vol. 2, 1931. See especially Sections 22, 29. 



