330 BELL SYSTEM TECHNICAL JOURNAL 



In this representation /(/) is regarded as the sum of a number of sinusoidal 

 components with fixed amplitudes but random phase angles. 



That the two different representations (2.8-1) and (2.8-6) of /(/) lead 

 to the same statistical properties is a consequence of the fact that they are 

 always used in such a way that the "central limit theorem*" may be used 

 in both cases. 



This theorem states that under certain general conditions, the distribu- 

 tion of the sum of N random vectors approaches a normal law (it may be 

 normal in several dimensions**) as iV— > co. In fact from this theorem it 

 appears that a representation such as 



If 

 /(/) = ^ (a„ cos Wnt + bn sin co„/) (2.8-6) 



where c„ and bn are independent random variables which take only the values 

 ± [w{fn)^ff'^, the probability of each value being h, will lead in the limit 

 to the same statistical properties of I(t) as do (2.8-1) and (2.8-6). 



2.9 The Normal Distribution in Several Variables^" 



Consider a random vector r in K dimensions. The distribution of this 

 vector may be specified by stating the distribution of the A' components, 

 Xi , X2 , • • • Xk , oi r. r is said to be normally distributed when the prob- 

 ability density function of the x^s is of the form 



(27r)-^^' I M r'^' exp [-WM-'x] (2.9-1) 



where the exponent is a quadratic form in the ic's. The square matrix M 

 is composed of the second moments of the .^•'s. 



M = 



Mil MI2 • • • Mi/c 

 where the second moments are defined by 



(2.9-2) 



Mil = '^'1 , Mi2 = .T1.T2 , etc. (2.9-3) 



I M I represents the determinant of M and x' is the row matrix 



x' = [.n , .T2 , . • • Xk\ (2.9-4) 



X is the column matrix obtained by transposing x'. 



* See section 2.10. 

 ** See section 2.9. 



2" H. Cramer, "Random Variables and Probability Distributions," Chap. X., Cambridge 

 Tract No. 36 (1937). 



