How has the assumption that N and the Xi are independent been used in Example D of Secti...

How has the assumption that N and the Xi are independent been used in Example D of Section 4.4.1?

Reference

Random Sums

This example introduces sums of the type

where N is a random variable with a finite expectation and the Xi are random variables that are independent of N and have the common mean E(X). Such sums arise in a variety of applications. An insurance company might receive N claims in a given period of time, and the amounts of the individual claims might be modeled as random variables X₁, X₂, . . . . The random variable N could denote the number of customers entering a store and Xi the expenditure of the ith customer, or N could denote the number of jobs in a single-server queue and X_i the service time for the ith job. For this last case, T is the time to serve all the jobs in the queue. According to