290 Expectation of Moments of Frequency Distributions 
If in equations (6) and (7) of Chapter I we replace /i^'*' by Xk'''^ '^"^^ /^[k, N] 
Xik. N] we find 
E [f^\r, m - l^u; mf^ = 1.3.5... {2 m - 1 ) (xi-i, a^j)'" ] 
+ 
^ m+i 
+ ... 
= 1.3.5... {2m + 1 ) l^i-^, ir?i (^f,, ^[3, m 
+ 
1 
Hence* 
^ A'] — /^[z-, AT]]" 
+ ... 
(8). 
-E" [/*'['■, m ~ f^['\ A']? 
1 fl ^ 
3 U 
I (9). 
-3.^![^!;>-(^;:Vp 
When r = 2, for the case in which the frequency distributions of all the variates 
follow the Laplace-Gauss law, we have 
2 / 1 {/) 
.(10). 
* Cf. Biomctrilxa, Vol. xii, pp. 162—163, equations (14), (15) and (16). 
