288 Expectation of Moments of Frequency DlstrihuUons 
It follows that if all the quantities m**' are equal to one another, then 
/"■"[)•, A'] = /^'[j-, A'], whatever the frequency distributions of the separate variates 
may be. If however the quantities mj'* are unequal then jv'] =^ /^'[r, w] ^nd, in 
general, is also not equal to Efi\,,^i^i^. 
On the other hand 
N r 
V X 
E^'l, ^] = 1 S i (- ly CV ^,.3 = (- 1)^ C," m;;_ mt,.,. 
A'] + 2; (- 1)'' 6V' m^^^ j^r^ rn^r-h. N] + {-^Y ' (r - 1) m ^ 
ii = \ ' ' 
When r = 1 we get 
SO that the quantities /u.'[i,,v] and //."[i, jv"] are identically equal. If in the relation 
(1) we put r equal to 2, 3, 4 successively, we find, 
^ ^' (i) ^ 
^>"[3, A^] = /^[3, A] + ^ S ' - A']] + \ .2^ ^ - ,v]]^' 
3^ ^ 
4 ^ 
Eil [4, A'] = "[4, iV] + ^ 2 - A ] J + 2^ .f*^ [^'h - "^[1. iVj] 
...(2). 
II 
(1) Noting that yu-%._ .yj is the arithmetic mean of the mutually independent 
quantities [A'',' — m^^^, \X.i — . . . \X' y - m^^ ''^ and that E [X/ — m'^\ = fi"' , 
3 m r€ 
1 ^ 
while ^[X/ — TO^^'^]*'' = /i.|'^|, we find from relations (3) of Chapter I, on replacing 
»«ti, N] by ^ /^t'' = f^b; N]> A] by ^ 2^ [/i,^!.' - (^|.'^)'] and so forth, 
1 ^ 
, , 1 m (m - 1 ) ,«-2 / 1 ^ (,■) \ 
^ [a^ A']]™ = Ml,, + ]^ — 2 — ^^'j ,ri " )■] j 
1 (7/lt-4] 
1 
/n-3 / 1 ^ p (,) (,) (/) ('')^3^\( , 
Cf. Biometrika, Vol. xii, p. 160, equation (6). 
