COMMUNICATION IN PRESENCE OF NOISE 73 



the probability density of .vo could be obtained directly from p{x, u) by 

 letting M -^ in (4-6). As it is, the a-'s appearing in the resulting expression 

 must be divided by r to obtain the correct expression. Thus, the probabiUty 

 of finding xo between .Tq and .vo + ^.Vo is 



which is of the x" type frequently encountered in statistical theory. 



It follows that the probability of finding u in {u, u -f du) and Xo in 

 (.To, .Vo + (/.Vo) at the same time is Pq{u, .Vq) du dxo where 



po(u, .Vo) = p(u, Xo)po{Xo) 

 1 /uxoY'-'-^'* ^ ^, .„., _,„^,„„^,,.„,, (4-11) 



4rr(iV + 1/2) 



(..,. \ Ar/2-1/4 



The replacement of (.v, «) in (4-6) by (m, .Vo) should be noted. 



Now that we have the probability density of u and .Vo we may combine it 

 with the probability (4-8) that all A' of .Ti, • • • , Xk exceed .Vo when Xq and u 

 are fixed. The result is the answer to the problem stated in Section 1 : 



Prob. (Pi(2, •••,Px<2>/^o0 



= / du \ dxnp^{u,x^\\ — /'(.Vo, 7^] 



Jo ♦'0 



This result is more complicated than it seems, for ^o(;/, .Vo) is given by (4-11) 

 and P{xi^, ii) is obtained by integrating p{x, u) of (4-6) from .t = to a; = x^ 

 in accordance with (4-7). The remaining portion of the paper is concerned 

 with obtaining an approximation to (4-12) which holds when N and K are 

 very large numbers. 



5. Behavior of Prob. {P\Q, • • • , PrQ > PoQ) as X and A' Become Large 



In this section we introduce a number of approximations which lead to a 

 manageable expression for Prob. (PiQ, • ■ ■ , PrQ > PoQ) when N and K 

 become large. 



Since u and Xo are sums of independent random variables, namely 



n=-N 



A'o = z^ B„ , 



the central limit theorem tells us that the probability density Po{u, Xo) ap- 

 proaches a two-dimensional normal distribution centered on the average 



