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BELL SYSTEM TECHNICAL JOURNAL 



Then, computed by means of the formulas derived in AppendLx B, Fig. 3.3 

 gives a curve of Re and a curve of Max P(R;b), each versus b. Since the 

 curve of Re cannot be read accurately at 6 ?5r; 1, there is included also a 

 curve of Rc/y/l — b-, from which Re can be accurately and easily com" 

 puted for any value of b; incidentally, the curve of Re/y/l — 6' is simul- 

 taneously a curve of -s/Telb, on account of (3.22). From Fig. ?i.7i it is 

 seen that Re varies greatly with b but that Max Pji-;^ varies only a little, 

 as also is seen from inspection of Figs. 3.1 and 3.2 giving curves of P{R\b) 

 as function of R with b as parameter. 



In Fig. }).?), the curve of Re shows that for 6 = 1 the maximum of P{R;b) 

 occurs ai R = 0; and the curve of Max P{R;b) shows that Max P{R;\) ^ 

 0.94, agreeing to two significant figures with the value 0.9376 found near 

 the end of Appendix B. - 



4. The Distribution Function for the Reciprocal of the Modulus 



At first, let R denote any real variate, and P{R) its distribution function. 

 Also let r denote the reciprocal of R, so that r = \/R; and let P{r) denote 

 the distribution function for r. Then -- 



P{r) = R'PiR) = P{R)/r\ 



(4.1) 



If P{R) depends on any parameters, P{r) will evidently depend on the 

 same parameters. 



The rest of this section deals with the case where W = R(cos + i sin 6) 

 is 'normal.' Since this case depends on 6 as a parameter, P(R) and P(r) 

 are here abbreviations for P{R;b) and P{r;b) respectively. 



As PiR;b) has the distribution function given by (3.4), the distribution 

 function for r will be 



P{r;b) = 



(Vl - b'-)r 



3 exp 



-1 



(1 - &VJ "L(i - b'yy 



(4.2) 



obtained from the right side of (3.4) by changing R to l/r and multiplying 



" For if r and R denote any two real variates that are functionally related, sa}- F{r, K) 

 = 0, and if dr and dR are corresponding small increments, then evidently 



P{r) \dr\ == P{R) \ dR \ whence 



Pir) 

 PiR) 



dR 

 dr 



bF/br 

 dF/dR 



In particular, if r = \/R, whence F = r — l/R, then (4.1) results immediately. 



For a somewhat ditYerent and more detailed treatment of change of the variable in 

 distribution functions, see Thorton C. Fry, "Probability and its Engineering Uses," 

 1928, pp. 1.S3-155. (Cases of more than one variate are treated on pp. 155-174 of the 

 same reference.) 



