ELEMENTARY SAMPLING THEORY FOR ENGINEERING 45 



the infinite universe case. Formula (12) does not give the same results 

 as (11) as it is most exact when njN is small and becomes absolutely 

 exact in the limiting case of an infinite universe where njN = 0. 

 This formula also approaches the Poisson Limit, in this case as X jN 

 approaches and w + 1 becomes infinite in such a way that the 

 product {n + \){X IN) remains constant and equal to a, say. 



The Poisson Limit, for the case of an infinite universe, was given 

 by Molina in the Appendix to the article in the Bell System Technical 

 Journal of January, 1924, already mentioned in this memorandum. 



Another point of interest is brought out when we note that in the 

 limiting form of (12) the Poisson gives us 



w{c, X) = X) —fr ' ^ = "at ' 



and for another pair of values of W and X ' 



t=c+l ^ • -'* 



Thus from properly chosen Poisson curves or tables we may obtain 

 the ratio Xi/X = ai/a which corresponds to the observed value of c 

 and the desired values of W and Wi. This ratio in exact formulae 

 is a function of N, n, and X also, but for many problems involving 

 small values of n/N and X jN the degree of approximation furnished 

 by this limiting form is fairly satisfactory and still further reduces 

 the amount of labor necessary in extending approximate results to 

 practice. 



The sort of procedure we have just been discussing may be facilitated 

 by means of a chart on which we show as abscissae values of c and 

 as ordinates values of the ratio of Xi/X which corresponds to various 

 values of W as shown by various curves and a specified value of Wi, 

 say Wi = .9. Such a chart would enable us to interpret roughly a 

 given Chart C for W = .9 in terms of other values of W. For precise 

 work this procedure is not to be recommended, and, therefore, no 

 charts of the nature just described are included herein. 



Approximations to the binomial other than the Poisson have been 

 discussed in many of the texts. In particular, for values of p in the 

 neighborhood of |, the well-known Laplace-Bernoulli integral 



1 r" 



-''dt 



will serve as an approximate value for Wi where the limits a and b 



