358 BELL S YSTEM TECH NIC A L JOURNA L 



Some of these considerations and relations have found application in 

 Section 5 in the evaluation of the cumulative distribution function for the 

 modulus R = I ir |. For this reason, the variate in the present section 

 will be denoted by R. though without thereby restricting R to denote the 

 modulus; rather, R will here denote any positive real variate, though it 

 should preferably be a 'reduced' variate, so as to be dimensionless, as in 

 equation (2.9). The restriction of R to positive values is imposed because 

 it is strongly conducive to simplicity and brevity of treatment, without 

 constituting an ultimate limitation. The reciprocal of R will be denoted 

 by r, as previously.^* 



We may wish to evaluate numerically the cumulative distribution func- 

 tion p{R'<R) = Q{R) or p{R'>R) = Q*{R) or both. Since these are not 

 independent, their sum being equal to unity, the evaluation of either one 

 determines the other, theoretically. However, when the evaluated one is 

 nearly equal to unity, the remaining one may perhaps not be evaluable 

 with sufficient accuracy (percentagewise) by subtracting the evaluated one 

 from unity. Then it would presumably be advantageous to introduce 

 for auxiliary purposes the variable r — 1/R, since evidently 



p(R'>R) = p{\/R'<l/R) = p{r'<r), (Dl) 



p(R'<R) = p{r'>r) = 1 - p{r'<r). (D2) 



Thus, if p{R'>R), in (Dl), is small compared to unity, it is presumably 

 evaluable with higher accuracy percentagewise by dealing with p{r'<r) 

 than with 1 — p{R'<R). Incidentally, after p{r' <r) has been evaluated, 

 it might be used in (D2) to arrive at a still more accurate value of p{R' <R) 

 than had originally been obtained directly by numerical integration. 



Assuming that we have a plot (or a table) of the distribution function 

 P{R), we can evidently evaluate 



P{R'<R') = / P{R)dR (D3) 



Jo 



directly by numerical integration, provided the plot is sufficiently extensive 

 to include R ; if not, we can, by (D2), resort to 



P(R'<R') = 1 - p(r'<r') = 1 - / P{r)dr, (D4) 



Jo 



assuming that a sulficiently extensive i)lot (or table) of P{r) is available 

 and applying numerical integration to it. 



Even if the plot of P{R) used in (D3) is sulficiently extensive to include 



'■• The restriction of R, and hence of r, to positive values is seen to be absent from equa- 

 tions (Dl), (D2), (D5) and (D6) but present in (D3), (D4), (D7) and (D8). 



