68 BELL SYSTEM TECHNICAL JOURNAL 



When the Z's are independent, so that Theorem 1 is applicable, 

 equations (94) and (95) respectively reduce to 



WP = E^' + 2i:^ft"^A. (98) 



\W\^ = Y. \ZrV + 2Re E ZkZ^, (99) 



although (96) and (97) remain unchanged. Thus, when the Z's are 

 independent, the following relations exist: 



W^ -!¥ =Y.(J? - Zr), (100) 



JW\^ -\W\' = Z (IXF - \Zr\'). (101) 



It is of interest to compare these with (90) and (91), which do not 

 require the Z's to be independent. 



When, further, not more than one of the Z's is of non-zero mean 

 value, so that at least » — 1 are of zero mean value, that is, when '^ 



Zr = 0, {r = 1, •••,i- l,i+ 1, ••-.«), (102) 



then (98) and (99) reduce to 



1^ = L ^. (103) 



TW = Z iXp- ^^^^^ 



After substitution of the value of W from the defining equation 

 (92), and with due regard to (102), equations (103) and (104), on 

 account of their importance and simplicity, may profitably be ex- 

 pressed in the form of two theorems, respectively, as follows: 



Theorem 4. // any number of complex chance-variables are inde- 

 pendent and if not more than one is of non-zero mean value, then the 

 mean of the squared value of their sum is equal to the sum of the means 

 of their individual squared values. 



That is, 



(Zi + • • • + Znf = Zi2 + . . • + Z„^ (105) 



provided the Z's are independent and not more than one is of non-zero 

 mean value, in accordance with (102). 



Theorem 5. // any number of complex chance-variables are inde- 

 pendent and if not more than one is of non-zero mean value, then the 

 mean of the squared magnitude (absolute value) of their sum is equal to 

 the sum of the means of their individual squared magnitudes. 



"An important practical instance in which one of the Z's is of non-zero mean 

 value will be found in connection with equation (120) in the problem treated m 

 Section 6. 



