148 Expectation of Moments of Frequency Distributions 
CHAPTER I 
I 
Consider a variable iiiagnitude X, admitting the values ^i, ^2, •••^it with 
probabilities j)], jx., ... pi,. Let us make N experiments, and suppose that the law 
of distribution of the values of the variable remains unaltered, and that the 
separate experiments are independent of one another. Denoting by Xi the value 
taken by the variable in the ith experiment and by ?(/ the number of times, out of 
the N experiments, that the variable X takes the value ^j, let 
"J ^ 
EX' - Ex,'- 
./=i 
: E{X ~ m,Y= E{X, - lu, )'■ =Xpj i^j 
.7 = 1 
M,., ,,v, = E [Z,,v, - m^Y 
We have, whatever be the law of distribution of the variable X 
Ic 
s Pi 
/,■ 
= ^ 1 
i=i 
i=i 
k 
7 = 1 
= N, 
'"l, (*) 
= iih, 
Ml, iN) 
0, 
A'o, (iV) 
= 1^0 = 
1. 
We find further, without difficulty : 
= //(o 
Ma 
= 7/i;, 
— 'ym.,111. 
+ 2//(i" 
= /»4 
Mr 
= vt.. 
- 
_i ?;ti + ...+(- 1)'' C,'' la.^n I'll'' + 
+ (- l)*--' C,'-' m.,nh'—' + (- 1 )'-' (r - 1) 
M/', (.v) = (A-i - '«,■-], LV) nil + ... + (- 1)'' G,'' in,._h^ (,v, wii'' + 
+ (- 1)'-- Gr' III., (.VI + (- 1)'-' {r - 1) mi'- 
.(1). 
...(2). 
