TM WO. 377 



Given the probability density of 9( ,, the probability density of the random 

 variable ^ can be estimatedc In other words, the aim is to estimate the 

 instantaneous aaiplitude of the wave at t = ti^ using the knowledge of the pro- 

 bability density of the phase angle » The method used here is discussed by Lee 

 (i960). 



The phase angle ^ need only be specified between and 2Tr. Let the 

 probability density of o^ be P^iy)i and that of '7 be?'.,(x). For every 

 value of 07 in its range (-A^A)^ there are two possible values of o< in its 

 range (0j2Tr)^ except when 'H ^ ±k^ 



Letting ^ X-/2, -t-f CjC^ (A.If9) 



equation (A=^) becomes; 



^ Z A^"^ 3 j (A-50) 



Where the range of b is ( JL|ti , ^^-fr, -^ ZTT )= The relation between the range 

 of variables of X and g of the raiidom variables "y and^ is thus; 



X - A ^•'^ 2 ; 



(A- 51) 



where 



2-= J2,ti-^y . (A-52) 



Assuming that S^ and S^ are two solutions of (A-51) for g in the interval 

 (Ji-ltt^^tti +> 2Tr )t then: 



PCK<'nCX-hiK)r (p6^.<:§^^'+^*)-H(P(*i<5*<:**^-J^}. (A.53) 



This equation states that an event of <» occurring in (x^ x+dx) is dependent 

 upon the event of § occtirring in (z, ^'-td^) and (*2^2--i4d*) and nowhere else. 

 If there were only one solution of (A=5l)^ only one term would appear on the 

 right side of (A=53)<. Equation (A=53) leads toi 



A-10 



