Probability Theory and Telephone Transmission 

 Engineering 



By RAY S. HOYT 



Part I of this paper contributes methods, theorems, formulas and graphs 

 to meet a previously unfilled need in dealing with certain types of two- 

 dimensional probability problems — especially those relating to alternating 

 current transmission systems and networks, in which the variables occur 

 naturally in complex form and thus are two-dimensional. The paper is 

 concerned particularly with "normal" probability functions (distribution 

 functions) in two dimensions, which are analogous to the familiar "normal" 

 probability functions in one-dimensional probability problems. It supplies 

 a comprehensive set of graphs for the probability that a "normal" complex 

 chance-variable deviates from its mean value by an amount whose magni- 

 tude (absolute value) exceeds any stated value; in other words, the proba- 

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

 the mean value in the plane of its "scatter-diagram," that is, in the complex 

 plane of the chance-variable. It gives a comprehensive treatment of the 

 distribution-parameters of the "normal " complex chance- variable, and con- 

 venient formulas for the necessary evaluation of these parameters. For use 

 in various portions of the paper, as well as for various possible outside uses, 

 it supplies a considerable number of formulas and theorems on "mean 

 values" ("expected values") of complex chance-variables. 



Part II of the paper makes application of Part I to some important 

 problems in telephone transmission systems and networks involving chance 

 irregularities of structure and hence requiring the application of probability 

 theory. 



Introduction 



IN telephone transmission engineering a frequent problem is that of 

 determining the effects of random manufacturing variations upon 

 the value of some characteristic (for instance, a transfer admit- 

 tance, or a driving-point impedance, or a current-ratio) of a trans- 

 mission system or network.^ In certain cases, such effects may be of 

 great or even controlling importance in the performance of the system 

 and hence must be fully taken into account when designing the system 

 and when making calculations for predicting its performance. 



For example, in a multi-pair telephone cable the crosstalk between 

 any two pairs is directly proportional to (strictly, a linear function 



1 Such problems have in the past been handled by various approximate methods 

 the most satisfactory of which for many purposes was that described in a paper by 

 George Crisson, entitled, "Irregularities in Loaded Telephone Circuits," published 

 in this Journal for October, 1925. The method given in the present paper, while 

 necessarily more involved than approximate methods, yields more precise results; 

 and this additional precision is expected to be of importance in practice. IVIoreover, 

 there has been an increasing need for a comprehensive paper covering the entire 

 ground, and it is hoped that the present paper meets this need to a measurable extent. 



In Crisson's paper references will be found to various engineers in the Bell System 

 who had previously contributed to specific probability problems of the type dealt 

 with in Part II of the present paper. 



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