286 Expectation of Moments of Frequency Distributions 
^•2*- + l, (-V) 
N 
N-\ N 2r-l 
t fiZ. + s s 2 
[2r + l]! 
(A) ('2) 
+ ... + 
2r-l-i-,-r2-. ..-)•, 
1 .3.5.. .(2r + l)r ! 
3 
+ lJV-)-+2 JV" 3 5-r 
S ... 2 2 2 
?, = 1 ?2 = /, + 1 7 ). = 1 + 1 >'i = 2 ;'.2 -- 
n- 
"- (/,) (/.,) (//■-,) (',•) 
= 1.3.5...(2r + l)j^^-ir^;;;,3M^3,^j 
+ 
J .[-2] r-2 J _[-3] )-3 
:5 0' /^[2, iV] ^[5, J\T ■•"18* f^l2, iV] '^LS. /*[4, 
4- -L? 
~ K 1 ' 
-4 3 
1 [-21 r- 
81*' /^[2, jv] ^L3, jV] -y 1^12, if] y .^^ f^2 H 
Putting r = 1, 2, ... we find f 
1 
1 
^3, (AO = ]yr2 /^I''- AH 
3 2 
1 
1 (1 * 
10 
.(7)*. 
.(8). 
IV 
Relations (6) and (7) show that as N increases ^^^T^Ji^ tends to the limit 
M. 
2, (N) 
1.3.5... (2r - 1) and '-^^zrr^^ tends to the limit zero provided that 
2, ov) 
2 /A.,' 
JV 
2 
i = l 
(i) 
f^2h-l 
2 fi, 
i = l 
^ - ^'"M2/i-l. AM 
[i\^/*[2,JVlP"-' 
tend, as iV increases, to the limit zero, for A = 2, 3, ... r. Hence, if 
2 = 1 
or 
f^[r, N] 
[N/^[2, N]Y 
* Cf. Biometi ika, Vol. xii, p. 154, equation (23). 
t Vide Tschuproff, " Zur Theorie der Stabilitat statistischer Reihen" (Skandinav.isk Akttiarietid- 
skrift, 1919, p. 82) and Biometiika, Vol. xii, p. 155, equation (26). 
