64 BELL SYSTEM TECHNICAL JOURNAL 



Formulas (28), (29), (30) involve z^ and \z\'^. To evaluate z^ we 

 square z and take the mean value of the result; to evaluate \z\'^ we 

 multiply s by its conjugate z and take the mean value of the result. 

 We thus obtain from (64) the formulas 



72 = j^2^ + • • • + &„V + . . . + 2h,h^t + • • •, (65) 



W= |&i|'|zi|'+ ••• + \hnV\zn\'+ ••• +2RehhtZ,Zt+ ••■, (66) 



where 5 = 1, • • • w — 1 and t = s -\- I, • • • n. These two formulas 

 can also be written 



? = ' L" irV + 2''i: Kbcl^t, (67) 



FF = 'l" |6rI'Ts^+ 2Re'E"&AiX, (68) 



T S<t 



corresponding respectively to formulas (94) and (95) in Subsection 4.3. 

 When the subscripted Z's are independent, and hence the sub- 

 scripted z's are independent, equations (65) and (66) respectively re- 

 duce to _ 



? = b,W + • • • + &n's„^ (69) 



|s|^ = \bA'\z,\'+ ••• + \bn\'\z„\', (70) 



on account of Theorem 1 in Subsection 4.1 together with equation (63), 



4. SOME FORMULAS AND THEOREMS ON MEAN VALUES OF 

 COMPLEX CHANCE-VARIABLES 



The present Section supplies a considerable number of formulas 

 and theorems on "mean values" ("expected values") " of complex 

 chance-variables. Many of these formulas and theorems have already 

 been used in Part I, and further use for them will be found in Part II; 

 while outside of this paper they may well find varied other uses. 



The theorems are word-statements of the simpler and more fre- 

 quently useful of the formulas ; the remaining formulas are more gen- 

 eral and are not simple enough to be profitably expressed as theorems. 



Theorems 1 and 2 regarding the mean of a product of complex 

 chance-variables and Theorem 3 regarding the mean of a sum are 

 generalizations of the corresponding known theorems for real chance- 

 variables, are formally the same as the latter, and are susceptible of 

 the same sort of proofs. These three theorems furnish a natural basis 

 for the remaining theorems, besides having extensive other uses. 



" Defined just after equation (2). 



