190 Expectation of Momentii of Frequency Dist7^ibntions 
(3 ) When ?(. = 3 and n = 4, we have similarly : 
W 
(3) 
r, (N) 
= Nv,,, (A^) + W{N-l) E [X, - Z(A.)]- [X, - 
(4) 
(^,) = iV^i'.., (A') + ^N{N-l)E[X,- X^^if^ [X, - X(^)Y 
- - ■ + %N{N -1){N-2)E [X, - [X, - X^^^nX.-- X^^^J 
+ N(N-1){N- 2) (iV- 3) E [X, - [A% - [X, - X,^)]*- [X, - X^^r^f. 
Determining W^J.^jj^r, and from these relations and substituting the 
values found in (10), we obtain E[v',.^^^r) — t-^.i^v)]^ and E[v'r,(N) ~ ^i-AmY- Their 
expression exhibits no special difficulties, but is so unwieldy that I do not 
give them here, contenting myself with the deduction of E [v',.^ — v^, (a^P and 
E[v',.^^N) — Vr,{N)Y which will be shown below (see Chapter IV, § ill). 
Ill 
(1) Noting that 
S [X,:-X(,v)]-^- S (X,:-/hO— X'[X(A^)-mJ-^= S (X,;-m,)^ 
1 = 1 
and putting 
i = l 
V^;\% = E^l^{X,-m,y 
2 (Xi-7ni) 
!=1 
r A^ 
Z 
■'{r, s) 
2, (A^) 
E\ 2 (X,-X(^))^ 
S (Xi - mO 
2 (Xi-mi) 
•(14), 
we 
find 
If'"'' =E 
2 (A i — X{N))' 
F^"" = E 
2, (A') 
2 {Xi-m,r 
(15), 
h 1 „(m-/i,2ft) 
or 
^^2, (N) = V,, (AO W (^^) ~ ^ ^^'"■('^^ 
•(16), 
7i = l 
and, on the other hand, 
i U 
r-l 
2.(iV) - 2, (AT) 
/i = 0 
r-l 
2,(A') 
A=0 
-ft, 
i\r/t ^2,(,N) 
"2^ (- 1)'^ c'r 4x c^':;/"^"' + (- n /....u^) . 
