(4) [50 marks] From past experience a professor knows that the number of marks that a student gets on their final examination is a r.v. with mean 75 (out of 100). Give an upper bound for the probability that a student's score will exceed 85. [5 marks] If we know that no student gets under 40 marks, improve the above upper-bound on the probability that a student's score will exceed 85. [5 marks] If we further know the variance in the student scores to be 25, use Chebyshev's inequality to improve the above upper-bound on the probability that a student's score will exceed 85. [5 marks] Using the one-sided Chebyshev inequality from Question (3), further improve the above upper-bound on the probability that a student's score will exceed 85. [5 marks] Under the same assumptions that the student scores have mean 75, variance 25 and are always greater than 40, define S, as the score of student i in a class of n students. Assume that the scores of all students are mutually independent. If Z = 0, prove that Z; [0, 1], E[Z;] = and Var[Z] 14 [10 marks] 60 Using the one-sided Chebyshev's inequality to bound the deviation of Z = Z; from its mean, calculate how many students would need to take the examination to ensure, with probability of at least 0.999, that the class average would be at most 85? [10 marks] Using the Chernoff bound to bound the deviation of Z = Z; from its mean,, calculate how many students would need to take the examination to ensure, with probability of at least 0.999, that the class average would be at most 85? (NOTE: exp(x) = e in the Chernoff bound) [10 marks]