The Zipf (n, α = 1) random variable X introduced in Problem 2.3.9 is often used to model the “popularity” of a collection of n objects. For example, a Web server can deliver one of n Web pages. The pages are numbered such that the page 1 is the most requested page, page 2 is the second most requested page, and so on. If page k is requested, then X = k.
To reduce external network traffic, an ISP gateway caches copies of the k most popular pages. Calculate, as a function of n for 1 ≤ n ≤ 106, how large k must be to ensure that the cache can deliver a page with probability 0.75.
A Zipf (n, α = 1) random variable X has PMF
The constant c(n) is set so tha Calculate c(n) for n = 1, 2,…, 6.