PROBABILITY THEORY AND TELEPHONE ENGINEERING 45 



b = — I respectively, the ellipses degenerate to superposed straight 

 line segments coinciding with the w-axis or the I'-axis respectively; 

 owing to this superposition of the straight line segments the "proba- 

 bility density" on the resulting straight line locus is not constant but 

 varies in accordance with the 1 -dimensional normal law, as expressed 

 by equation (1). 



With the object of reducing the number of parameters from 2 to 1 

 and of dealing with variables that are independent of units, it will be 

 preferable not to deal directly with the original chance-variable 

 tu = u -^ iv, which is referred to the central principal axes ucv, but 

 rather to deal with the "reduced" chance-variable W = U + iV de- 

 fined by the equation 



W = w/S = u/S -f iv/S = U + iV, 



(15) 



which is referred to the central principal axes Z7CF coinciding with the 

 central principal axes ucv (Fig. 1), so that the position of any point T 



♦ V 



I 

 I 



W--U + IV 



Fig. 1 



in the TF-plane will be represented by TF = U -{- iV. Thus we shall 

 be directly concerned with the scatter-diagram of W = U -{- iV in- 

 stead of with that of w = m -f iv. 



From (12) it is easily found that the probability law, say Pu,Vt 

 for W = U + iV{s 



Pu,v — 



1 



xVl - b^ 



exp 



IP V 



1 -f & \ -h 



(16) 



which contains only the one parameter &, defined by (13), while more- 

 over the variables V and F are independent of units. Thus the 

 "reduced" complex chance- variable W = U -\- iV given by (15) is 

 defined as "normal" if its probability law can by the proper choice 

 of a pair of rectangular axes UCV in the plane of its scatter-diagram 

 be written in the form (16); the ?7CF-axes are the "central principal 



