Exercise 1. Fix n 2 1. Let X ~ Unif[0, 1] and let Y ~ Binomial(n,X). Find the expectation 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 an intuitive explanation for this.) Exercise 2. Let N be a random variable taking nonnegative integer values, and let X1, . . . , 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 an alternative proof of EM = ElNllElX] using the formula My\") = MNUOE MXGD- Exercise 3. Let X,: ~ Geom(q) be iid, and let N ~ Geom(p). Show that the MGF of Y = 2111 X. is _ pqe' My\") _ 1- (1 -pq)e" What distribution is this? Exercise 4 (Fun exercise, suggested by Clark Lyons). A fair -sided die is rolled until a 6 appears. Find the expected number of rolls, conditioned on the event that only even numbers are rolled. Hint: The answer is not 3. An important observation is that the information \"only even numbers are rolled\" makes it likely that there were very few rolls. So we should expect that the expected number of rolls conditioned on the event is strictly less than 3