COMMUNICATION IN PRESENCE OF NOISE 71 



This behavior follows from the fact that the probability density of Pk has 

 spherical symmetry about the origin (because all the .4^ 's have the same 

 a). For the probability that Xk is less than some given value x is the prob- 

 ability that Pk lies within a sphere of radius .t^''' centered on Q, and this, 

 because of the symmetry, depends only on .v and the distance «^'- of Q from 

 the origin. Accordingly, we write p{x, u)dx for the probability that 

 X < Xk < X + dx when the a'„'s (and hence n) are fixed. 

 The probability density p{x, u) may be obtained from its characteristic 

 function: 



dz 



r JL. 



(4-5) 



p{x, u) = {lir) ' / e '"""[ave. e''^] 



J— 00 



[A- -1 



iz 2^ y'n 

 n =—N 



= IT ave. exp [izy'n] = (1 — 2iz)~^~^'~ e.xp [ms(l — 2/3)"^] 

 where we have used (4-3) and, since y„ is distributed normally about Vn, 

 ave. exp [tzy„\ = {Itt) \ e "^ - djn 



= (1 — 2is)~^'" exp [y'ni^i'^ — 2/z)~'] 



Hence 



= (2x)"' f (1 - 2/c)-'''-''- exp fi3«(l - lizV' - izxldz 



(4-6) 



pix,u) =(2x)"' /* (1 - lizy-"- exp [izuil - lizT' - hx\ dz 



O-l/ / \2V/2-l/4 J- r/ \l/2i -(«+x)/2 



where it is to be understood that .v is never negative. The Bessel function 

 of imaginary argument appears when we change the variable of integra- 

 tion from z to / by means of 1 — 2iz = 2t/x, and bend the path of integra- 

 tion to the left in the / plane (6). This expression for the probability density 

 of the sum of the squares of a number of normal variates having the same 

 standard deviation but different averages has been given by R. A. Fisher 

 (7). 



We are now in a position to solve the following problem which is somewhat 

 simpler than the one stated in Section 1: Given the 2X -\- 1 coordinates 

 AI'' oi the point Pq and the 2A" -\- 1 numbers J5„ so that the coordinates 

 A n -f- Bn of the point Q are given. What is the probability that none of the 

 K points Pi, P2, ■ ■ ■ , Pk, whose coordinates A [ are drawn at random from 

 a universe distributed normally about zero with standard deviation a = 1, 

 be inside the sphere centered on the given point Q and passing through the 

 other given point Po? In other words, what is the probability that all K of the 



