Al. a. TCHOITPROFF 
149 
Conversely, expressing the quantities in terms of fx,., we find 
>•-! ^ 
my = E [{X - ?»,) + Will'' = + i, C/' ?»,'-'' /x/, + fx, 
h = 2 
r-1 
ii-iy w,_„ m,„ ,.v, = s (- ly cv* I 
... (8). 
II 
(1) Noting that 
" X 
SX; 
i = l 
we write | S A"",- 
in the form 
tis ... 
7' ; 
x^p. 
where, as is known, the summation with regard to j extends from 1 to the smaller 
of the two integers r and N; the summation with regard to i.,, ... ij extends to 
all integral and unequal values of , ?„, ... ij from 1 to X, and the last summation 
extends to all positive integer or zero values of r^, ... satisfying the rehition 
ri + r„ + ... + Vj = r. 
Passing to mathematical expectations we find hence 
m,. u\) = EX 
rl 
ri ! ! ... r,- : 
,EX,p EX;p...EXi/. 
T ! 
where the summation with regard to j extends to all positive integer values from 
1 to the smaller of the two numbers r and X, while the second sum extends to 
all positive integer values of , ?-„, ... Vj, satisfying the relation : 
7\ + 7-2+ ... + ? J = r. 
If r "> X, we have consequently : 
14), 
i\ j=i 
where the R,.j coefficients are independent of X and are defined by the relation 
1 r! 
Ri-^j 
HI 
r, m,^_...m,^, 
1.2.3 ...j rj r,!...?-! 
and the summation extends over the ranges specified above. 
