PROBABILITY THEORY AND TELEPHONE ENGINEERING 65 



4.1. Mean of a Product of Independent Complex Chance- Variables 



The following Theorems 1 and 2 relating to the mean of a product 

 of complex chance-variables are very important notwithstanding their 

 limitation to chance-variables which are independent. 



Two discrete chance-variables are said to be "independent" (or 

 " uncorrelated " or "non-correlated") if the probability that either 

 takes any given value is independent of the value taken by the other. 



Two continuous chance-variables are said to be "independent" if 

 the probability that either lies close to any given value is independent 

 of the value taken by the other. 



Theorem 1. // any number of complex chance-variables are inde- 

 pendent, the mean of their product is equal to the product of their indi- 

 vidual means. 



That is, if the Z's are independent, 



Z1Z2 ••• Z„ = ZiZ2 ••• Z„. (71) 



Theorem 2. If the magnitudes {absolute values) of any number of 

 complex chance-variables are independent, the mean of the magnitude of 

 the product of these complex chance-variables is equal to the product 

 of the means of their individual magnitudes. 



That is, if the |Z| 's are independent, 



IZ1Z2 •••Z„| = IZ1IIZ2I ••• \Zn\. (72) 



For the validity of Theorem 2 it is not necessary that the angles 

 of the chance-variables be independent, but only their magnitudes. 

 Moreover, if ^i, • • • 0„ denote the angles of Zi, • • • Z„ and $ the angle 

 of their product, then, by Theorem 3, 



5 = ^ + • • • + ^, (72a) 



whether or not the 0's are independent. 



4.2. Mean of a Sum of Complex Chance-Variables 

 The following Theorem 3 is of unlimited scope, in the sense that it 



involves no assumption as to independence of the chance-variables. 

 Theorem 3. Given any number of complex chance-variables, which 



need not be independent, the mean of their sum is equal to the sum of their 



individual means. 



That is, whether or not the Z's are independent, 



Zi + • . . + Z„ = Zi + . . . + Z„. (73) 



