PROBABILITY THEORY AND TELEPHONE ENGINEERING 41 



is 1-dimensional) is defined as "normal" if its probability law, or 

 distribution function, can by the proper choice of orip;in be written in 

 the form 



where, by definition of the term "probability law," Pndu represents 

 in general the probability that the unknown value n' of a random 

 sample consisting of a single value of the chance-variable lies between 

 u and u + du; or, what is ultimately equivalent, the probability that 

 u' lies in the differential range u db du/2, namely in the differential 

 range du containing the point u. Su is a distribution-parameter called 

 the "standard deviation" of u and defined by the equation 



ti^Pudu, (2) 



■00 



the superbar connoting the "mean value," or "mean," of any chance- 

 variable to which it is applied. In this paper the term "mean value" 

 is used as an alternative for "expected value," namely the "weighted 

 average value" with the w€ighting in accordance with the probability 

 of occurrence of each particular possible value of the variable. (Sec- 

 tion 4 supplies a considerable number of formulas and theorems on 

 mean values of complex chance-variables — and hence of real chance- 

 variables, by specialization.) 



From the foregoing definitions, it is easily verified that 



r 



t/ — 



Pudu = 1, 



which corresponds to taking unity as the measure of certainty. 



It will be recognized that the chance-variable « in equation (1) is 

 related to the original given chance-variable, which will be denoted 

 by X, by the equation u = x — x. Hence w = 0, as has already ap- 

 peared in equation (2); thus the origin is at the "center" c of the 

 distribution, namely the point 7ic with respect to which as origin the 

 "mean value" of the chance- variable is zero, that is, such that u — Uc 

 = 0, whence Uc = ii = 0. If, in terms of the original variable x, the 

 position of c is denoted by Xc, then x — Xc = and hence Xc = x. 

 Since u — x — x, it is seen from (2) that 



SJ = (x - x)^ = .V. (3) 



The probability that the magnitude (absolute value) [ u' \ of a ran- 



