102 BELL SYSTEM TECHXICAL JOURNAL 



This is obtained by integration by parts using 



\Mien av >> 1 but 1 << a - r, Bennett has shown that (3.10-17) 

 leads to 



r . ^ ^ ( -^ Y' 1 



/ p{u) ail ~ \ - — I exp 



Jo \27ra/ a - V ^ 



(v - af 



( 



1 - 



3(fl + v)~ — 4i'" 

 8av{a — v)- 



(3.10-18) 



Fig. 6 — Probability density of envelope R of I{t) = P cos /)^ + /_v 



This formula may also be obtained by putting the asymptotic expansion 

 (3.10-19) for p(v) in (3.10-17), integrating by parts t\\ice, and neglecting 

 higher order terms. 



Wben av becomes large we may replace Io{av) b}' its asj'mptotic expres- 

 sion. The expression for p(v) is then 



Thus when either a becomes large or v is far out on the tail of the probability 

 density curve, the distribution behaves like a normal law. In terms of the 

 original quantities, the normal law has an average of P and a standard devia- 

 tion of 1^0 "• This standard deviation is the same as the standard deviation 



