PROBABILITY THEORY AND TELEPHONE ENGINEERING 67 

 (88) and (89) become 



ZiZr- ZrY- = ZiZr'- Zr), (90) 



E \Zr-Zr\' = E{\Zr\'- | Z, | ') , (91) 



where each summation J2 covers the set r — \, • • • n. 



4.3. Mean of a Squared Sum of Complex Chance-Variables 



With a view to arriving at Theorems 4 and 5 below, and also several 

 formulas which are more general than the theorems but are not simple 

 enough to be profitably expressed as theorems, let Zi, • • • Z„ denote 

 any complex chance-variables; and for brevity let IF denote their sum, 

 so that 



TF = Zi + • • • + Z„. (92) 



As indicated by its title, this Subsection will be concerned particu- 

 larly with formulas for IP and | W\^, but it will also include formulas 



for W' and \W\\ 



Squaring IF, given by (92), and then applying Theorem 3 gives 



PF = E Z.2 + 2 E E ZuZ,, (93) 



r=l h=l k=h+l 



or, in a briefer notation. 



^ = E Z.2 + 2 E ZhZ,, (94) 



the second E in (94) thus denoting double summation. ^^ 



Taking the product of W and its conjugate W and then applying 

 Theorem 3 gives 



\W\' = Z |Z.|2-f 2ReEZ,Zfc. (95) 



Applying Theorem 3 to (92) and then squaring the result gives 



w' = ZZl +2Z Z.Zl. (96) 



Taking the product of W and W gives 



\W = i: |Z;|' + 2ReEZ,Z,. (97) 



^^ In (95), • • • (99) the summations evidently cover the same sets of values as in 

 (94). 



