Probability Theory and Telephone Transmission Engineering

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 magnitude (absolute value) exceeds any stated value; in other words, the probability 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 convenient 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.