160 Expectation of Moments of Frequency Distributions 
Substituting in (2), we find : 
1 (wJ-^l 
(2) Efx'iV^ may also be calculated by means of recurrence formulae. Put 
N 
1=1 
Am) 
EN'" fju',"' = E 
From relation (8) of the first chapter we find : 
.(7). 
.(8). 
Noting that v''^^'^^^^^ = Nfx,., we find hence : 
>; (A ) 
,(3) 
(■1) 
'r. (N) 
and, in general, 
im) 
f, (.V) 
2 m-'^D 
()») 
i = l 
...(9), 
.(10), 
where 
'% 1 
— ^Hir 
)•, 
— h'-r 
r, i 
h = i-\ 
r, )ii - 
1 ^ ' )•,»!- 3 i;m-2 ^ 
n 2 ,, ,, in— 2 
— <^)n i"2r 
l)^,,/Li,'"-= 
' r,m-'2 
\ 
r, (ill,) 
'«-3 •> . i 
= /^3r 2 C; ./i,->(m,-3-j)r + 3C,^^/io,= 
i=o 
and so on. 
Substituting in (10), wc obtain 
V 
im) 
r, (.V) 
r, (iV) 
+ iY[-.— '4] 2 , ./i,.'>„._3-y,,. + 3C,„V.,,>,— 4 + ...'^ ...(12). 
I ./■=<» J j 
m - 1 /j I 
