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Exercise 1. Fix n 2 1. Let X ~ Unif[0, 1] and let Y ~ Binomial(n,X). Find theexpectation and the variance of Y. Do you recognize the distribution of Y? (Answer:it turns out that Y is the discrete uniform distribution on {0, l, . . .,n}. Try to find anintuitive explanation for this.) Exercise 2. Let N be a random variable taking nonnegative integer values, and letX1, . . . , XN be independent identically distributed (i.i.d.) random variables. Let Y =X1 + + XN, and let ]E[X] denote the common expectation of the X;. Give analternative proof ofEM = ElNllElX]using the formulaMy“) = MNUOE MXGD- Exercise 3. Let X,: ~ Geom(q) be iid, and let N ~ Geom(p). Show that the MGF ofY = 2111 X. is _ pqe‘ My“) _ 1- (1 -pq)e" What distribution is this? Exercise 4 (Fun exercise, suggested by Clark Lyons). A fair ﬁ-sided die is rolled untila 6 appears. Find the expected number of rolls, conditioned on the event that onlyeven numbers are rolled. Hint: The answer is not 3. An important observation is that the information “onlyeven numbers are rolled” makes it likely that there were very few rolls. So we shouldexpect that the expected number of rolls conditioned on the event is strictly less than 3.