OBSERVED FLUCTUATION 



161 



via) = P'ia) = 



(lP{a) 

 da 



(4) 



is the fractional density of members of the set at a. 

 In other words, p{d(Ja is the fraction of signals with 

 amplitudes between a and a + da. The function p{a) 

 is often called the differential distribution function. 

 Analogous concepts can be formed for intensity and 

 level distributions which have here been sketched for 

 amplitude distributions. If we call the intensity dis- 

 tributions Q{I) and q{I) (the capital denoting again 



70 



95 



97 

 98 



Figure 2. Cumulative distribution of the amplitudes 

 of 50 signals. 



the integrated and the lower-case symbol the differ- 

 ential distribution), and likewise the level distribu- 

 tions W{L) and w{L), then the integrated distribu- 

 tion functions are, of course, related to each other 

 very simply by the equation 



W{L) 



= wUo log - j 



P(a) =0(7) =0—, (5) 

 2pc 



since the fraction of signals with an intensity less 

 than I is identical with the fraction of signals having 

 an amplitude less than a if a and I are related to each 

 other by means of equation (1). For the differential 

 distributions it follows that 



,_ dW(L) dP(a) da ^ da 



since the amplitude and level are related by equation 

 (2); thus, 



a 



Similarly 



dQ(I) dP{a)da PC 



(l{J) = —77- = -7— 77 = V{a)~ (7) 



dl da dl a 



because the amplitude and intensity are related by 

 equation (1). 



Distribution functions can be determined experi- 

 mentally, and the limiting distribution function will 

 be approximated by the experimentally found distri- 

 bution more and more closely as the size of the sample 



40 

 50 

 60 



75 



90 



95 



98 



o 



o 



o 



o 



Q 



O 



o 







o 



o 



o 

 s 



Q 



o — 



o 



(t)' 



Figure 3. Cumulative distribution of the amplitudes 

 of 287 signals. 



is increased. On the other hand, distribution func- 

 tions can also be predicted theoretically by assuming 

 that fluctuation is caused by certain assumed mech- 

 anisms. Figures 2 and 3 show two integrated distri- 

 bution functions which were obtained from actual 

 samples. One of these samples is plotted on proba- 

 bility paper, on which any Gaussian distribution be- 

 comes a straight line."^ 



Two theoretically predicted distribution functions 

 will be discussed here. The first of these is the so- 

 called Rayleigh distribution. Let us consider, as an 



w{L) = p{a) 



8.69 



(6) 



•• A Gaussian distribution is one in which the density p(a) 

 is given by the function 



1 -(.a-my'/2i 



A Gaussian distribution will usually result if a large number of 

 random processes aflect the value of the argument a. The two 

 parameters Oo and S are the average value and the standard 

 deviation of a respectively. 



