3. [12 marks] i) [6 marks] Suppose that U1, U2, ... are iid uniform on (0, 1). a) [3 marks] Find the limiting distribution of M = n(1 max{U, , Un}). b) [3 marks] Verify your answer above by estimating (cdf and/or pdf) of M100 using computer simulation. Do not forget to include any figures in your submission (either em- bedded within your typeset document, or submitted as a separate file named "Q3"). ii) [6 marks] Suppose that (X) is a sequence of pairwise independent random variables such that P(Xn = n) = 1/n = 1 P(Xn = 0). a) Find the expectation lim, EX, and the limit of (Xn) in probability. b) What is the reason lim, EX and the limit of (Xn) in probability are not the same? Explain using mathematical concepts covered during lectures.