Al. a. Tchouproff 
141 
"frequencies which would become established in an indefinitely great number of 
occurrences " often fails to make the very statement of the problem clear to the 
reader, and occasionally it would appear, to the author: when reading published 
papers one not infrequently feels that the author does not give himself a full 
account as to what it is he is really calculating. 
Little harm follows so long as the problems dealt with are comparatively 
simple. But at the present time there are problems waiting for solution which 
are so complex that the slightest obscurity in their formulation threatens t(j 
become a source of error in the final deductions. 
When we start with " mathematical probability " and " mathematical expecta- 
tion " as a foundation we substantially simplify the mathematical exposition. The 
logical analysis of the conclusions to which we are led is not injuriously affected 
by the substitution of one set of terms for the other during the calculations. 
(2) If the variable magnitude X can take the values ^j, fj, ... with proba- 
bilities ^^1, p-2,, ... jJfc, I call the system of values ^i, ■■• ?it f-nd the values 
p-i, ... pic associated with them "the law of distribution of the values of the 
variable X." The law of distribution of values lies at the base of empirical 
"frequency curves," just as the mathematical probability of an event lies at the 
base of its statistically established frequency. 
Denoting by the symbol EX the mathematical expectation of the variable 
magnitude X, we have as is well knoAvn : 
EX=ipi^;, 
1=1 
h- 
where ^ Pi= 1- 
( = 1 
I call the variable magnitudes X, Y, Z, ... mutually independent, if the law cjf 
distribution of each of them remains one and the same whatever values are given 
to the others. In this case EX remains constant for all possible values of the 
variables F, Z, 
If the law of distribution of A" does not remain the same for different values of 
y, Z, the variables X, Y, Z, ... are mutually dependent. The mathematical 
expectation of the variable A' on the supposition that Y has received the value 
Z the value etc., I denote by ■ •^A' and call it the " conditiunul mathe- 
matical expectation of A' on the supposition that the remaining variables have 
received definite values." 
It follows from the definitions that 
E {X + Y + Z + ...) = EX + EY-^ EZ+... 
both in the case when the variables are mutually independent, and when they are 
correlated, and that 
E{XYZ...) = {EX )(EY)iEZ)..., 
if the variables are mutually independent. 
