Al. a. Tchouproff 
295 
'•[1, N]} 
and hence 
= — i^r-- - ^ J + ^ -iv^- ]^ ]" 
■ 2 1^ ,,) 4 1 ^ u, (0 
~ ^-'^ ' i? ^2 ^"^1 -™[i,iv]]- 
, , fl ^ (0 ('•) U. 
+ 4 .-^ ^3 - /ip, iv] iv]|*t- 
* Fide Tschuproff, " Zur Theorie der Stabilitiit statistischer Reihen " {Skandinaviak Aktuarie- 
tidskrift, 1919, p. 85, equation (4)) aud Biometrika, Vol. xii, p. 193, equation (23). 
t The problems discussed above can also, as regards most of them, be attacked by the methods 
employed in the fourth chapter of the first part of my paper " On the Mathematical Expectation of the 
Moments of Frequency Distributions" {Biometrika, Vol. xii, pp. 194 — 210). Suppose that the aggregate 
of all the possible values that may be assumed by .Yj, X,, ... X^v reduces to fi , $1' ••• Denote by 
PjM the probability that the variate A'j may take the value Jy, where pj{i) = 0, if fy does not occur among 
the possible values of A'^; denote by Hy the number of times, that among the N experiments performed, 
one or other of the variates takes the value Jy. 
Putting ft- = 4 2 pj(i); ■ ■ 
we find 
k k 
m,(i) = S ijy(0 fy'- ; my., .VI = ^ P} i/ ; 
3=1 J=l 
" Vi^^ [fy - '"i<''']'' ; iKr, iV] = 4 ^ ^ p,,C) [?y - m^^)Y : 
iv i=iy=i 
\ k Ik 
= V 2 ny?y=»i'[i,,v], where ;»,'[,., ,vj = tt S 
iv /=i iv y=i 
1 * 
M [>•, M = AT 2 «; [fy - '"[1, iV]]'" ; 
IV y=i 
1 k r 
''V,(A')=„ M>Ky-A'(,vi]'-= S ( -l)''C,,''(^t'V, .v])'■-''w^ where = A'lAr, - 
iv - 
Of the various quantities discussed above, fi'i,-, .vi is the only one that cannot be expressed in terms 
of fij and Jy. The others can be studied by means of relations analogous to those numbered (1) to (6) in 
the fourth chapter of the first part of my paper (Biometrika, Vol. xii, pp. 195 — 6). This method is how- 
ever of little practical use, as the resulting formulae are too cumbrous. Thus with Eiij — Npj, we get 
for the mathematical expectation of the square of rij, 
., ,. E7ij'2 = mpj^ + N2}j- 2 (2)yW)2 
i=l 
= mpp + Npj (1 - py) - f [p/) -pjf. 
For the mathematical exjsectation of the cube of iij we find 
£«/ = N3/>/ + 32^i)/ (1 -pj) + Npj [1 - Spj + 2pf] - 3 [Npj - 2pj+l] 2 [p^O -pjT + 2 2 [y/) -_pyP 
and so forth. 
19—2 
