Chapter 5.3, Problem 186E

The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise.

186. [T] Complete sampling with replacement, sometimes called the coupon collector ‘s problem, is phrased as follows: Suppose you have N unique items in a bin. At each step, an item is chosen at random, identified, and put back in the bin. The problem asks what is the expected number of steps E(N) that it takes to draw each unique item at least once. It turns out that $E\left(N\right)=N,H_N=N\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{...}+\frac{1}{N}\right)$ . Find E(N) for N = 10. 20. and 50.