Probability Functions for the Modulus and Angle of the 

 Normal Complex Variate 



By RAY S. HOYT 



This paper deals mainly with various 'distribution functions' and 'cumulative 

 distribution functions' pertaining to the modulus and to the angle of the 'normal' 

 comy)lex variate, for the case where the mean value of this variate is zero. Also, 

 for auxiliary uses chiefly, the distribution function pertaining to the recijirocal 

 of the modulus is included. For all of these various probability functions the 

 paper derives convenient general formulas, and for four of the functions it supplies 

 comprehensive sets of curves; furthur, it gives a table of computed values of the 

 cumulative distribution function for the modulus, serving to verify the values 

 computed by a difTerent method in an earlier paper by the same author.^ 



Introduction 



IN THE 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 ana- 

 lytical 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 F); 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 rec- 

 tangular components (X and F); also, its two polar components, radius 

 vector and vectorial angle (R = \ Z \ and 6). 



This paper deals mainly with the modulus and the angle of the complex 

 variate,^ which are often of greater theoretical interest and practical im- 



'"Probabihty Theory and Telephone Transmission Engineering," Bell System Tech- 

 nical Journal, January 1933, which will hereafter be referred to merely as the "1933 

 paper". 



' Throughout the paper, I have used the term 'complex variate' for any 2-dimensional 

 variate, because of the nature of the contemplated applications indicated in the first 



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