PROBABILITY THEORY AND TELEPHONE ENGINEERING 47 



Moreover, 



PbiR^ <R' < R2) = piiR' > Ri) - pb{R' > R^) (18) 



= p,{R' <R2) -poiR' <Ri), (19) 



where Ri and R2 denote any two stated values of R such that Ri < R2. 



From (13) the total possible range of b is seen to be from — 1 to 

 + 1, corresponding to the total possible range of 5„/5u from <» to 0, 

 with b = corresponding to S^/Su = 1. However, it will evidently 

 suffice to consider for b the range to 1, corresponding to the range 

 1 to for S,./Su, which will be secured by choosing 5„ as the greater 

 and hence S^ as the smaller of the two "standard deviations" (with 

 the «a'-axes chosen correspondingly, of course). 



The graphs in Figs. 2 and 3 show the relation between R and 

 pb{R' > R) with b as parameter; similarly, Figs. 4 and 5 show the rela- 

 tion between R and the quantity pb,o{R' > R) defined by the equation 



Pb,o{R' > R) = pb{R' > R) - Po{R' > R). 



(20) 



Here po{R' > R), being a particular value of pbiR' > R), plays the 

 part of a reference value. It is a natural reference value, being the 

 value for 6 = 0; and it can be evaluated immediately and accurately, 

 since its exact formula is merely 



PoiR' > i?) = exp ( - i?2). 



(21) 



.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 



R 



Fig. 2 



