Hi please use the given information to answer each part of the problem and please include a short explanation, drawing, or steps/formulas to improve my understanding of the problem. Thank you!

Zipf random variable . The Zipf r.v. is used to model the popularity of items such as books or videos in online libraries. The Zipf r.v. Z takes values k E {1,2,...,m} with pmf k'\" m p k = , wherea>0,andc= 16\". z( ) C E The interpretation is that if the videos in an online library are numbered 1 through m; in order of decreasing popularity, then a random request to View a video would be for video A: with probability p209). For the following questions let m = 2000 and a = 0.8 (suitable for a typical video library). (a) Plot the pmf of Z. (b) What is P{Z S 500}? (c) What is the mean and variance of Z? (d) What happens to the mean and variance as m ) 00 and a = 2? Hint: 220:1 k2 = \".2 6