Volume XII 
NOVEMBER, 1919 
NOS. 3 AND 4 
ON THE MATHEMATICAL EXPECTATION OF THE 
MOMENTS OF FREQUENCY DISTRIBUTIONS. 
By professor AL. A. TCHOUPROFF of Petrograd. 
CHAPTER III 
I 
DEC 23 191b 
(1) Let us put 
1 ^ 
2^ 2 [Xi — A(JV) ]*■ = Vr, {N) • 
.(1). 
We have v^^ (^vi = 0. 
Noting that E (Xi - Z,^,)'' = E [Xj - X^^r^Y, 
we find vr, m = E[Xi-X <.v, r = E [X, - Z,v, ]r. 
Replacing Zj — X by (Xj — mj) — (X — fth), we find 
V,., iN) = f^r + 't (- 1 )■' E {X, - m,y-i (Z(jv) - m,)i + (- 1 ya,, (jv) . 
i=i 
But [Z(^) - = ^ [(Z, - mO + (A - 1) (Z(^_i, - m,)?', 
where 
Hence 
1 
A(iv^_i) = — T 2 Xi 
A-l. = 2 
7j = 0 
(JV) 
+ (- 1 X I,, j/., + ^2 0/ (A - 1 (,v- 1) + ( A - 1 )'■ /.,, 1) 
)--2 
A- 
)--2 
Mr+ 2 (-1)'' 6'/ /Ar-A/^A,(iV-l) + (-!)'■ /i)-,(iV^-l) 
/t = 2 
= i (- ly^ 6'/ 
(2). 
/i=0 
On the other hand, replacing X^ — Z(jv) by 
i [(A- 1) A, - (A- 1) Z(^_i,J = [Z, - Z(.v_i)], 
we find : 
(iV) 
A-l\'- 
A 
2 (-1)'' CV »*/,,(JV^_l) 
Biometrika xii 
.(3). 
13 
