142 Expectation of Moments of Freqaencjj Distributions 
111 the case in which A' and }^ are correlated we have: 
J- 
EXY= ^ pi^iE^^'^V, 
/=i 
II 
(1) In investigations in the theory of probability we frequently have to deal 
with expressions of the type : N {N - I) {N - '2) ...{N - k + 1). Following the 
example of Capelli*, I use the slightly modified notation: 
N {N -\){N -2)... {N - k + 1) = m-^^ 
N{N + l){N+2)...{K+k-l) = i\^t+^-] 
.(1). 
•(2). 
Let N"^ i a^siiVt-^l 
./=(i 
The coefficients I have denoted by a, ^ are beginning to play an important 
part in the theory of finite differences^* and are of the first importance in all 
investigations into the law of large numbers. Their pi'operties were first studied 
systematically in Chapter III of Cramp's well-known ^no\\<., Analyse des refractions 
astru)to)niques et terrestres; some of their properties were discovered by investi- 
gators studying Bernoulli's numbers ; recently they have received the attention of 
the Italian mathematical school associated with Cesaro and Capelli. The methods 
I employ to solve fundamental problems of mathematical statistics are directly 
founded on certain properties of the a, jB coefficients. In view of the fact that I 
shall later on frequently make use of these methods, I state here, without proof, 
those properties of the coefficients a and 13 that I shall have to quote in the 
present paper |. 
(2) We have : 
= 1 ] 
«fc,*=l [ (3), 
ah/, = ^«^•-],,■+ j 
17273^:7^ = 1 . 27 37.; i ■ ■ 
* Vide Cnpelli : " Instituzione di analisi " and the same author's "L' analisi algebiica e T iuter- 
pretazione fattoriale delle potenza." {Giuniale di mati'rruitica di Dattaijlini, Vol. xxxi.) 
t Cf. A. A. Markoff, Calculus of Finite Differences (2iid edition). 
X Readers interested in the proofs of ttiese properties, many of them established for the first time by 
myself, will find a complete analysis in my paper in the Proceedings of the Petrograd Polytechnic 
Institute. 
