46 BELL SYSTEM TECHNICAL JOURNAL 



axes" of the scatter-diagram oi W = U + iV; and the "mean value" 

 of W is then zero, that is, T^ = 0. 



1.3. Graphs for the Probability of the Deviation of a Normal Complex 

 Chance- Variable from its Mean Value 



Before taking up the technical description of the graphs presented 

 in this Subsection, some indication of their field for practical use will be 

 furnished by the statement that the chance-variable w = u -{■ iv of the 

 next paragraph may, for instance, be identified with the chance- 

 variable h given by equation (II) of the Introduction, in case h is 

 "normal" and is of zero "mean value," so that h = 0; in case h 9^ 0, 

 then w would be identified with // — Ji. On referring to equation (II), 

 it will be seen that h there denotes the deviation of any transmission 

 characteristic from its nominal value; more generally, h may be any 

 complex chance-variable which is "normal" — or approximately "nor- 

 mal." 



The graphs here to be presented and described relate directly to the 

 "reduced" complex chance-variable W = U -\r iV given by equation 

 (15) in terms of the original chance- variable w = w + i'v and the 

 parameter 5 defined by equation (14). Assuming w to be "normal" 

 and of zero "mean value" {w = 0), it has the probability law formu- 

 lated by equation (12); and hence W = U -\r iV \s normal and of zero 

 mean value (W = Q), and has the probability law formulated by (16), 

 with the parameter b defined by (13). 



With W denoting the unknown value of a random sample consisting 

 of a single value of the chance-variable W, the graphs herewith rep- 

 resent the probability that the magnitude R' = \W'\ oi W exceeds ^ 

 any stated value R; that is, the probability that W lies without a 

 circle of radius R whose center coincides with the center C (Fig. 1) 

 of the scatter-diagram of W, so that the center of the circle is at 

 PF = 0. This probability will be denoted by pi{R' > R), the sub- 

 script b implying dependence on the parameter b. The complemen- 

 tary probability will be denoted by pb{R' < R) ', this is of course the 

 probability that R' is less than the stated value R; or, w^hat is equiva- 

 lent, the probability that W lies within a circle of radius R centered 

 at C. Of course the sum of the two foregoing probabilities is unity, 

 that is, 



Pt{R' > R) +pb{R' <i?) = 1. (17) 



^ In engineering applications it is usually preferable to deal with the relatively 

 small probability of exceeding, rather than with the complementary probability, 

 nearly equal to unity, of being less than a preassigned rather large value of R. 



