72 BELL SYSTEM TECHNICAL JOURNAL 



independent random variables .vi, xo, ■ • • , Xk will exceed the given value 

 0^0 when ii has the value defined by (4-4) together with the given values of 

 the An^^^ and BnS>} The variables .ti, X2, • • • , Xr have the probability- 

 density p{x, n) shown in (4-6) and .vo is defined by (1-2) and the given values 

 of the ^„'s. 



The answer to the above problem follows at once when we note that the 

 probability of any one of xi, • • • , Xk, say .Vi for example, being less than 



Xo IS 



' p{x, u) dx. (4-7) 







The probability of Xi exceeding xo is then 1 — P(xo, u) and the probability 

 of all A' of Xi, • ■ • , Xk exceeding Xo is 



[1 - P(xo, u)]'' (4-8) 



Instead of being assigned quantities, Xo and u are actually random varia- 

 bles when we consider the problem of Section 1. Now we take up the problem 

 of finding the probability density of u when xo is fixed. Thus, from (4-4), 

 we wish to find the probability density of 



«= i: u':' + bS' (4-9) 



n=—N 



in which the 2.V -f- 1 numbers An are drawn at random from a universe 

 distributed normally about zero with standard deviation a — \ and the 

 numbers B-n, • • • , Bq, ■ • ■ , Bn are given. It is seen that u is the sum of 

 the squares of 2N + 1 normal variates all having the standard deviation 

 0" = 1. The n\h variate, .4^"^ + Bn, has the average value i?„. This is just 

 the problem which was encountered at the beginning of this section. Equa- 

 tion (4-9) is of the same form as (4-3) and we have the following correspond- 

 ence: 



Equation (4-3) Equation {4-9) 



Xk u 



yn Air + Bn 



% ^ Bn 



n = Zlyn Xo = Z^B'n 



The probability that u lies in the interval u, u + du when .vo is given is there- 

 fore p{u, Xo) du where p{u, Xo) is obtained by putting u for x and xq for u 

 in the probability density p(x, u). 



Until now xo has been fixed. At this stage we regard B-n, • • - , Bq, • • • , Bn 

 as random variables drawn from a normal universe of average zero and 

 standard deviation i> = ar^^- — r^'-. If the standard deviation were unity, 



