Probability Functions for the Modulus and Angle of the Normal Complex Variate

I N T H E solution of problems relating to alternating current networks and transmission systems by means of the usual complex quantity method, any deviation of any quantity from its reference value is naturally a complex quantity, in general. If, further, the deviation is of a random nature and hence is variable in a random sense, then it constitutes a 'complex random variable,' or a 'complex variate,' the word 'variate' here meaning the same as 'random variable' (or 'chance variable'--though, on the whole, 'random variable' seems preferable to 'chance variable' and is more widely used). Although a complex variate may be regarded formally as a single analytical entity, denotable by a single letter (as Z), nevertheless it has two analytical constituents, or components: for instance, its real and imaginary constituents (X and Y); also, its modulus and amplitude (Z and 6). Correspondingly, a complex variate can be represented geometrically by a single geometrical entity, namely a plane vector, but this, in turn, has two geometrical components, or constituents: for instance, its two rectangular components (X and F); also, its two polar components, radius vector and vectorial angle (R = Z and 9). This paper deals mainly with the modulus and the angle of the complex variate, 2 which are often of greater theoretical interest and practical im1 "Probability Theory and Telephone Transmission Engineering," Bell System Technical Journal, January 1933, which will hereafter be referred to merely as the "1933 paper".