PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 325 



Su and Sv are distribution parameters called the 'standard deviations' of 

 w and V respectively. If / stands for u and for v generically, then 



P(t) = -7^ 



vfe,^-^;]' <'•'' ^' = /_j'^«'"- P.3) 



From the viewpoint of the scatter diagram, the distribution function 

 Pin,v) is, in general, equal to the 'areal probability density' at the point 

 u,v in the plane of the scatter diagram, so that the probabihty of falling 

 in a differential element of area dA containing the point ti,v is equal to 

 P{u,v)dA ; similarly, P{;u) and P{v) are equal to the component probability 

 densities. In particular, the probability density is 'normal' when P{u,v) 

 is given by (2.1). 



Geometrically, equation (2.1) evidently represents a surface, the normal 

 'probability surface,' situated above the u, r-plane; and P{u, v) is the ordinate 

 from any point u,v in the u,v-p\a.ne to the probability surface. 



The M,T'-axes described above will be recognized as being the 'principal 

 central axes,' namely that pair of rectangular axs which have their origin 

 at the 'center' of the scatter diagram of s = x + iy and hence at the center 

 of the scatter diagram of u> — u -\- iv, so that w = 0, and are so oriented 

 in the scatter diagram that m; = (whereas 2^0 and xy 9^ 0, in general). 



In equation (2.1), which has been adopted above as the analytical basis 

 for defining the 'normal' complex variate, the distribution parameters are 

 Su and Sv ; and they occur symmetrically there, which is evidently natural 

 and is desirable for purposes of definition. Henceforth, however, it will be 

 preferable to adopt as the distribution parameters the quantities S and b 

 defined by the pair of equations 



S' = Sl + Sl , (2.4) bS' = Sl - S; , (2.5) 



whence 



, __ Su Sy _ 1 [Sy/Su) ,r. ,,. 



»Jm "r Sy 1 -\- {Sy/SuJ 



From (2.4), S is seen to be a sort of 'resultant standard deviation.' The 

 last form of (2.6) shows clearly that the total possible range of b is 



— l^b^l, corresponding to '^^Sy/Su^O. 



The pair of simultaneous equations (2.4) and (2.5) give 

 2Sl = {\ + b)S-, (2.7) 2^; = (1-^.)^-, (2.8) 



which will be used below in deriving (2.11). 



'Equations (2.4) and (2.6) are respectivelj- (14) and (13) of my 1933 paper. 



