Al. a. Tchouproff 
195 
we have : 
N 
k 
= 2 7lj, 
.;=i 
k 
2 pj 
= 1, 
i = i 
/,- 
= 2 Pi ^j'' ; m/ 
i=i 
k 
t ' 
K 
= 2 JO; [^i — Vli 
A- 
l*" ; a/ = t pj' i - m 
j = l 
tC 
= ^ Pj ii = ™i' 
i = i 
But, 
EiH 
= i\>i 
Eni 
= i\^^j9r + ^j;,(l 
-Pd 
= Npi + 3iV[-2] p 
= N'pi + m^pi (1 - Pi) + Npi (1 - ^- ) (1 - 2/^-:) K 1 ), 
Enf = Np:,+im-^'^Pi^ + Qm-''^p>i + m-'^Pi' 
- + 6i\r^pr ( 1 - J^, ) + pr ( 1 - JO, ) ( 7 - 1 l^i) 
+ Np;(l-pi)(l - 6p; + 6pr) 
and in general, as is not difficult to see*, 
E7if= i m-narjp/= 2 mp.yS {-\y a.^n+f^n^fjp/ (2). 
/=i ft=i /=o 
Further, denoting by P/, the probability of iii taking the value h, and by 
-E^^''' , the conditional mathematical expectation of on the assumption that Ui 
takes the value h, we find : 
Pn =Glp,"(l-pd"-'\ 
eV ^(N-h)^P^, 
"j ' 1- Pi 
P 
Emnj = 1 PjJtE^^^ = 'i\iY-h)hG\p/^{l-pir--"-^Pi = N(F-l)piPi. 
Similarly we obtain : 
^n,- «,-^H,3 ^N(N-l){N-2)pi^pij)i^ I ^^^^ 
Eni^ni.^...n,^ = N{N-l) ...{N-k+l)pi^p!,...piJ ' 
* See my paper, " On the Mathematical Expectation of a Positive Integral Power of the Difference 
between the Frequency and the Probability of an Event," in the Proceedings of the Petrograd Poly- 
technic Institute. 
