196 Expectation of Moments of Frequenctj Distributions 
Enr n.j = iVt--' Pi pj + N^-''^ prpj 
= N'pcpj + N'' piPj (1 - Zpi) - Npipj (1 - 2j9i) 
En^^ nf ^N^-'^pi Pj + m-'^ pi pj (pi + + ^-'^ pi" pf 
= N'prpf + N'pipj{pi+pj-6pipj)+N-pipj(l-Spi-Spj + Tilpipj) 
- NpiPj (1 - 2pi - 2pj + 6pipj) 
En^7ij ^Ni-^piPj + 3Ni-'^prpj + m-^prpj [ (4), 
- N'pr Pj + SN-'p;'pj (1 - 2pi) 
+ N'piPj (1 - 9pi + Upf) - Npipjil - 6pi + Qpi') 
Enr 11 j «/,. = N^~''^pipjp,, + N'^''^prj)jPh 
= pi'pj ph + N^piPjPk (1 - 'opi) - N''pipjpi, (3 - Wpi) 
+ 2Npipjpk(l-dpi) J 
r, r., 
.(5). 
/i, = l /lo = l 
and in the general case : 
7i, = l /(.2 = 1 /'k = ^ 
(6). 
If the numbers Vi^, vi^, ... iiij^, referred to k series of independent experiments, 
then we shoiild have : 
• • • = J^K' ■ • ■ ^i^ik" 
/i, = l h, = l hk = l 
(2) Passing from the mathematical expectations of the numbers of repetitions 
to the mathematical expectations of frequencies, we find : 
Epi' =p, 
Epr=pf + j^Pi(l -pi) 
Epi' = + ^Pi' (1 -pi) + ^^ Pi (1 - Pi) (1 - 2^;) 
Epi' = Pi' + IrPm - Qpi) + ^ Pi' (1 -Pi) a - npi) 
+ ^sPi (1 -Pi) (1 - % + 6pr) 
Epi'r^S ^,^r" S (-iyar,r-n+fl3r.k^.fjPi^ 
in 
•(8), 
