PROBABILITY THEORY AND TELEPHONE ENGINEERING 39 



("expected values") of complex chance-variables. These formulas 

 and theorems hnd frequent and important uses in the present paper; 

 and outside of the paper they may well find varied uses. 



The method of treatment characterizing the present paper will now 

 be very briefly indicated in the remainder of this Introduction. 



As a preliminary step toward this objective we shall now return to 

 the functional formulas (II) and (III) with the remark that, if the 

 K's and ^'s were all real quantities and if these formulas were such 

 that // also were a real quantity, then the "direct" problem would be 

 to calculate the probability that // would lie within any stated linear 

 range, say ha to hb; thus the probability problem would then be one- 

 dimensional, and the well-known existing probability theory for real 

 quantities would be immediately applicable, including the correspond- 

 ing known methods and formulas for evaluation of the distribution- 

 parameters. 



When, as in .the present paper, the K's and ^'s are in general com- 

 plex quantities, the corresponding probability problem is inherently 

 two-dimensional. The distribution-parameters, which naturally are 

 more numerous than in the one-dimensional case, could be evaluated 

 in a roundabout way by an extensive process of resolution into rectan- 

 gular components; but it is believed that very superior advantages are 

 possessed by the probability methods and formulas contributed by the 

 present paper, for dealing with complex chance-variables in a more 

 direct manner, as set forth at some length in Sections 2 and 3, exten- 

 sively utilizing Section 4. The advantages of this method for evalu- 

 ating the distribution-parameters are perhaps particularly marked 

 whenever there is involved a summation of propagated effects, as in 

 transmission lines; for then, as will appear more concretely in the 

 applications in Part II, the necessary summations can be accomplished 

 much more easily and the resulting expressions are much more com- 

 pact and manageable than if a method employing rectangular resolu- 

 tions were used. 



Regardless of which method is used for evaluating the distribution- 

 parameters, the new material contributed by Subsection 1.3 is neces- 

 sary for the complete numerical solution of the problem in any specific 

 case where the "resulting" complex chance-variable // is "normal." 

 It may be recalled that this will be the case when h is a linear function 

 of the ^'s and the ^'s themselves are "normal." Even when these 

 two conditions are rather far from being fulfilled, however, it is known 

 from certain rather broad theoretical considerations that in many 

 practical problems h will be approximately "normal"; it may perhaps 

 be recalled that one of the most important among a set of sufficient 



