PROBABILITY THEORY AND TELEPHONE ENGINEERING 55 



manner, conformably to the implicit definition in the preceding para- 

 graph, we could proceed in a purely analytical manner, as outlined in 

 Subsection 2.2 below. However, in recognition of the very substantial 

 aid to thought and description furnished by the concept of the "scat- 

 ter diagram," for graphically representing any two-dimensional dis- 

 tribution, this concept will here be invoked in framing the definitions 

 and in deriving the desired formulas. 



Proceeding on this basis, it will be found that three of the "leading 

 distribution-parameters" are certain "average values" pertaining to 

 the scatter-diagram of the contemplated chance-variable; for any 

 "average value" pertaining to the scatter-diagram is equal to the cor- 

 responding "mean value" ("expected value") pertaining to the chance- 

 variable, when the "mean value" is defined as just after equation (2). 

 It will be recalled that there a superbar applied to the symbol denoting 

 any chance-variable was used to connote the "mean value" of the 

 chance-variable. In the present Section (2) , owing to the above-noted 

 relation, the superbar may interchangeably be regarded as connoting 

 either an "average value" pertaining to the scatter-diagram or the 

 corresponding "mean value" pertaining to the chance-variable. 



Having in mind the definition of the "scatter diagram" of any 

 complex chance-variable Z = X -\- lY, let XAY (Fig. 10) designate 



Z=X + i Y 

 z= X + i y 



Fig. 10 



the pair of rectangular axes with respect to which the scatter-diagram 

 of Z is plotted, A designating the origin of the XA F-axes. Also, let 

 T designate any plotted point in the scatter-diagram; and let c desig- 

 nate the "center" of the scatter-diagram, namely the point w hose 

 position Zc with respect to the XA F-axes is such that Z — Zc = 0, 

 whence Zc = Z. Further let xcy designate a pair of axes through c 

 parallel to the XA F-axes, and ucv any other pair of rectangular axes 

 through c; and let w = w -f w represent the position of the point T 

 with respect to the wCT-axes, the position of T with respect to the 

 a:c3;-axes being represented by z = x -\- iy and with respect to the 



