ELEMENTARY SAMPLING THEORY FOR ENGINEERING 43 

 Recalling that 



log(l +X) =x-1a^2 + 1x^_1x^+ ...^ 



which converges for X- < 1, 

 we have 



iog.(i. + i.x) = iog(i+i)--g|:|(^)', 



whence 



log F{N, n, X, /) = log ( 1 + ^^) - log ( 1 + ^) 



+ ZZ-. X. -EZ, tM -EZ^ 



Neglecting terms of the second order in 1/w, 1/A^ and 1/(A^ — «), 

 we have as an approximation 



1 mK! vA- \ (t'-^t .X^-2Xt±l±X-t X-'-X-2\ 

 \ogF{N,n,Xj)=--^—-^ -^—^ j. 



If now we select as our value of /, / = (nlN)X, we have log F{N, 

 n, X, t) = {X - 2)I2N which is > 0; if X > 2, 



F{N, n, X, t) = g(^-2)/-w, 



which gives us as an approximate value for the maximum term irt, 



where / = {n/N)X, 



^■-[-^)[n)['-n) ""-""■ '■'^ 



Having this term, it is a simple matter to calculate the other terms 

 necessary for evaluating W{c, X) by means of the exact equations 



^'+^-'''iv /+1 'N-n-X-^tJ' ^^^ 



( t N-n-X + t-\ \ 



TTt-l = TTi I — ; • 1 . (9) 



\fi — t -\~ 2 X — t -\- 2 / 



