ON THE MATHEMATICAL EXPECTATION OF THE 
MOMENTS OF FREQUENCY DISTRIBUTIONS. 
: PART 11. 
By professor AL. A. TCHOUPROFF of Petiograd*. 
CHAPTER I 
.... ^ I . ^ ' 
Let Xj, X2, ... be N variates each following Us otvn law of frequency 
distribution and let N experiments be performed, — the first on the variate X^, the 
second on the variate X^, and so forth, the last on the variate Xj^, where the iV^ 
experiments are mutually independent. Put : 
EXi = m^^'^; EXf = m./>; EXi^ = m/^; 
E {Xi - = m/^ - [m,<'''p = /^o"'' ; E [Z,- - r»i<'']'' = /a,"'' ; ■ 
\ N IN 
S^mi<'» =TO(i,jv]; ^ S m,<*' =m[,.,jvr]; 
\ N 1 * . ' 
Let Xl be the chance value obtained by the experiment on the variate X,:, 
and put further 
1 N 
^tX! = X(y), 
^^\X) = (A ) ; E [X(A') - nh^ (y,Y = fir, (itf). 
I N 
Noting, that mj, (a.) = xr 2 ^Z/ = , 
-iV j = i 
we see that 
f 
/i,., (iv) = ^ [Z(ivr) - jvi]*" = S (-1)'' CV (iv) . 
Conversely, 
r 
W/-, (iV) = ^ [Z(j^) - m[i, + vnfi, 2V]]'' = 2 C/'y»''[j,,v]/i^_/,,(Ar,. 
/)=(! 
* This paper was most kindly translated by Dr L. Isserlis from the Russian and read by him with 
Professor Tchouprofif's permission to the Society of Biometricians and Mathematical Statisticians, 
June 7, 1921. 
