2. (Credits for Graduation) Answer the following under the assumption that the average BU student has taken 140 credits by graduation (this includes BA and BA/MS students). (a) Knowing only this average, use Markov inequality to find the best possible upper bound for the fraction of BU students graduating with at least 160 credits. Make sure to define the random variable that you use in the application of Markov inequality. (Ignore all practical limits, i.e., assume there is no upper bound on the number of credits a student may obtain.) ~ (b) Demonstrate that this is a best-possible bound by giving a distribution for which this bound holds with equality. (Hint: consider a distribution of the form k. X for some X Bernoulli(p) for an appropriate k and p.) (c) Now use the fact that each BU student needs at least 128 credits to graduate. Improve your bound from (a) by applying Markov inequality to a different (but related) random variable. (d) Using this new fact from (c), how does this change your solution to (b)? (e) Now you further learn that the standard deviation of the total credits per graduating student is 6. What is the Chebyshev bound on the fraction of students who can graduate with at least 160 credits?