1. (5 + 4 + 4 = 13 marks) a) Consider a Poisson-distributed Random vari- able N with mean A = 4. Determine the probability that N is Within one standard deviation of the mean (including the endpoints). You can either calculate the probabilities directly from the probability mass formula for the Poisson, or if you wish, you may take the probability pN(4) = Prob(N = 4) = e_4(4)4/4! = 0.1954 and then calculate the other probabilities you need recursively using pNUi) 2 gm\": 1) * 4/13; I: = 5, 6,... and pNUi) = pN(k+1)*(k+1)/4; k=3, 4,.... b) What does Chebyshev's inequality tell you is the upper bound on the prob- ability that N is more than one standard deviation from its mean? What does this tell you about the tightness of Chebyshev's inequality? 0) Now, consider this Poisson random variable to be the sum of 4 i.i.d. Pois- son random variables {Ni}; i = 1,2,...,4, each with mean 1. Apply the Central Limit Theorem to the sum N 2 23:1 N,- to approximate the prob- ability that N is Within one standard deviation of the mean. Compare and comment With the answer from part a). N.B. Prob(1 g Z 3 1) = 0.6826