PROBABILITY THEORY AND TELEPHONE ENGINEERING 43 



If, with a view to generalizing (5), we inquire as to the probabiHty 

 P(\u' — Uo\ < r) that ii' deviates from any fixed value no of u by an 

 amount whose magnitude is less than any stated value r, and if now 

 we let r' and ro denote \u' — Uo\ and \iio\ respectively and R, R\ Rq 

 denote r/Su, r'/Su, ro/Su respectively, then 



pi\u' - uo\ <r) = p{R' < R) 



~ 2 L \ \2 / \ V2 



When Uq = this formula correctly reduces to (5). 



erf/^ii^^-erf^^"-^ 



(11) 



1.2. The Normal Complex Chance- Variable 

 Before proceeding to the "normal" complex chance-variable it 

 should be remarked that, although any 2-dimensional chance-variable 

 can be represented either as a complex chance-variable z = x -\- iy 

 = lj.exp{ir]) or as a pair of real chance- variables (x,y) or {fj.,r}), 

 nevertheless the two modes of representation, though of course mu- 

 tually equivalent, are not always equally advantageous. For the 

 types of problems contemplated in the present paper, the complex 

 representation has important advantages resulting from the fact that 

 the chance-variable when so represented is formally a single entity 

 and subject to the laws and transformations of complex algebra. In 

 Sections 2, 3, 4 of Part I and also in Part II, the complex representa- 

 tion possesses very great advantages. In the present Subsection, how- 

 ever, which is mainly concerned with formulations of the 2-dimensional 

 "normal" probability law (distribution function), the representation 

 in terms of a pair of real variables is the more advantageous. In this 

 Subsection, therefore, the complex representation is used only in those 

 places where it is particularly conducive to brevity and sharpness of 

 statement, and to simplicity and clearness of correlation with the 

 remainder of the paper where the complex representation is mainly 

 used. 



The normal complex chance-variable (which of course is 2-dimen- 

 sional) may be defined in several mutually-equivalent ways. Here a 

 complex chance-variable z will be defined as "normal" if its proba- 

 bility law can, by the proper choice of a pair of rectangular axes u,v 

 in the plane of the "scatter-diagram" of s, be written in the form 



1 / «2 ^,2 \ 



u and V being the pair of coordinates of any point of the scatter- 



