40 BELL SYSTEM TECHNICAL JOURNAL 



conditions for approximate "normality" is that the ^'s be numerous 

 (w a large number). 



As stated in the Synopsis, Part II makes application of Part I to 

 some important problems in telephone transmission systems and net- 

 works involving chance irregularities in structure. One of these prob- 

 lems, namely that in Section 5, is the general problem already outlined 

 in connection with the equations in this Introduction, 



Part I : Theory 



1. PROBABILITY OF THE DEVIATION OF A NORMAL COMPLEX 

 CHANCE-VARIABLE FROM ITS MEAN VALUE 



Toward the end of the Introduction it was stated that in many 

 problems of the types contemplated in the present paper the distri- 

 bution of the "resulting" complex chance-variable is approximately 

 "normal." 



To meet a previously unfilled need in the solution of such problems, 

 this Section of the paper supplies (in Subsection 1.3) a comprehensive 

 set of graphs for the probability that a "normal" complex chance- 

 variable deviates from its mean value by an amount whose magnitude 

 (absolute value) exceeds any stated value; that is, the probability that 

 the chance-variable lies without any specified circle centered at the 

 mean value in the plane of its "scatter-diagram." These graphs are 

 accompanied by sufficient explanation to enable them to be understood 

 and used without any necessity for studying the formulas from which 

 they were computed — which, because of their length and complexity, 

 have not been included in this paper.^ 



To furnish the necessary precise basis for the graphs, Subsection 1.3 

 describing them is preceded by Subsection 1.2 giving analytical defini- 

 tions of the normal complex chance-variable and its distribution- 

 parameters; and these quantities are discussed at moderate length 

 there. 



To lead up to the normal complex chance-variable, it is preceded 

 by a brief review of the normal real chance-variable (Subsection 1.1), 

 which is more familiar. 



1.1. The Normal Real Chance-Variable 



In order to lead up to the normal complex chance-variable (which 



is 2-dimensional) it will be recalled that a real chance-variable (which 



3 The formulas are given (with derivations) in an unpublished Appendix (Ap- 

 pendix A). Another unpublished Appendix (B) gives various concepts and definitions 

 employed in two-dimensional probability theory, and also gives various analytical 

 and graphical ways of representing probability. Still another (C) treats a problem 

 of crosstalk in a telephone cable. 



