PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 319 



portance than the real and imaginary' constituents. The modulus variate 

 and the angle variate, individually and jointly, are of considerable the- 

 oretical interest; while the modulus variate is also of very considerable 

 practical importance, and the angle variate may conceivably become of 

 some practical importance. 



The paper is concerned chiefly with the 'distribution functions'^ and the 

 'cumulative distribution functions' pertaining to the modulus (Sections 3 

 and 5) and to the angle (Sections 6 and 7) of the 'normal' complex variate, 

 for the case where the mean value of this variate is zero. The distribution 

 function for the reciprocal of the modulus is also included (Section 4). 



The term 'probability function' is used in this paper generically to include 

 'distribution function' and 'cumulative distribution function.' 



To avoid all except short digressions, some of the derivation work has 

 been placed in appendices, of which there are four. These may be found 

 of some intrinsic interest, besides faciUtating the understanding of the 

 paper. 



1. Distribution Function and Cumulative Distribution Function 

 IN General: Deeinitions, Terminology, Notation, Relations, 



AND Formulas 



The present section constitutes a generic basis for the rest of the paper. 



Let T denote any complex variate, and let p and a denote any pair of 

 real quantities determining r and determined by t. (For instance, p and 

 (7 might be the real and imaginary components of r, or they might be the 

 modulus and angle of t.) Geometrically, p and a may be pictured as gen- 

 eral curvilinear coordinates in a plane, as indicated by Fig. 1.1. 



Let T denote the unknown value of a random sample consisting of a 

 single r-variate, and p' and a' the corresponding unknown values of the 

 constituents of r'. 



Further, let G(p, a) denote the 'areal probability density' at any point 

 p,a- in the p,(7-plane, so that G(p,a)dA gives the probability that t falls 

 in a differential area dA containing the point r; and so that the integral of 



paragraph of the Introduction, and also because the present paper is a sort of sequel to 

 my 1933 paper, where the term 'complex variate' (or rather, 'complex chance-variable') 

 was used throughout since there it seemed clearly to be the best term, on account of the 

 field of applications contemplated and the specific applications given as illustrations. 

 However, for wider usage the term 'bivariate' might be preferred because of its prevalence 

 in the field of Mathematical Statistics; and therefore the paper should be read with this 

 alternative in view. 



^The term 'distribution function' is used with the same meaning in this paper as in 

 my 1933 paper, although there the term ' probability law' was used much more frequently 

 than 'distribution function,' but with the same meaning. 



