188 Expectation of Moments of Frequency Distributions 
IN 
Putting Z(A--2)= AT_ 9 
N 
we have 
[X, - [X, - Z(A.)] = 1) (Z, - mO - (Z, - m,) 
-(X- 2)(Z,A._2) - m.)] [(iV - 1 ) (X, - mO - (Z, - m,) -{N-2) (Z(^_2) - 
/AT"— 9\2 /iV— 2\'^ 
= /(Z„Z,)- [(Z, - + (Z,-m,)] [Z(.v_2,-m,] + ( j [Z(^_2) 
where 
/(Z, , Z,) = [X, - m,] [Z, - 77h] ~ [(Z, - m,) + (Z^ - m,)]^. 
Hence : 
^[Z,-Z(.v)]'-[Z,-Z(,v)]'- . ' 
=^^i/(z„z,)]'-+ i)''C';:^{/(z,z,)}'-'' (^y' 
X ([(Zi - + (Z2 - wO] [-^ (iv-2) - - [Z(^_2) - mi]^l^ 
Z 
+ i 2 {-\^^G\G'^-H^')\uA^-^^E{f{X,,X,^^^^^^ 
But (see Chapter I (15) and (16)) 
Bnt.(|)-i ^ , 
^{/(Z„ Z,)h' =1: I (- 1)/ 6'{ 
/=0 9' = 0 iV 
^ {f{X„ Z,)]-^ [(Z, - + {X, - m,)?"-* 
2A+J/-/f . / (Z- 1/ 
= 2 Z (- 1)-^C , C.^, 2„_^ /J.,^.h+f~k-gf^s-h-f+g- 
f=0 5 = 0 iV ■ 
Hence : 
N{N-l)E[X,-X(^^^nX,-X^K^r=i % (- l>^ ^^~y < C f,r,f-,^r-f+, 
+ C',.' 2 S (- 1>' j^y^^r^^ C;^._j C.^^H^.,+f.g flr.i-f+g 
* The development of £{/ (A'l A'a) [(Xj - wij) + (A'o - HiijP''-*^ in powers of N does not contain 
any terms of higher degree than while at the same time the development of /ij. ^jf^.^) contains no 
terms of higher degree than - . To obtain Elv^, ^fff-'''r,^N)^^ correct to l/N* it is con- 
sequently sufficient to carry our calculations as far as k = 2t. 
