5. Consider a simple random walk on the set {0, 1, 2, ..., N} in which each step is to the right with probability p or to the left with probability q = 1- p. Absorbing barriers are placed at 0 and N. Show that the number X of positive steps of the walk before absorption satisfies E(X) = {Dk-k+ N(1 − Pk)} where D is the mean number of steps until absorption and pk is the probability of absorption at 0.

2. Suppose that X and Y are independent random variables, both with the standard normal distribution i.e. X, Y~ N(0, 1). For pe [-1, 1], define Z:=√1-p²X + pY. Show that E[Z|Y] =pY.

Suppose that a random sample of sizen is taken from a Poisson distribution for which the value of the mean e is unknown, and the prior distribution of e is a gamma distribution for which the mean is Po. Show that the mean of the posterior distribution of e will be a weighted average having the form Y,X, + (1– Yn)Ho, and show that yn →1 as n- *.

Assume that A, B, and C are subsets of a sample space S with Pr(A) = 0.8, Pr(B) = 0.6, Pr(C) = 0.15. 1. Find Pr(A'), Pr(B'), and Pr(C'). Pr(A') = Pr(B') = Pr(C') = 2. If Pr(A U B) = 0.85., find Pr(AN B). Pr(AN B) = 3. Suppose that we know that B and C are disjoint events. Find Pr[BUC. Pr[BUC] = 4. Suppose that instead of being disjoint C C B. Find Pr[B U C]. Pr[BUC] =

2. Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X1, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞ Hint: to lower bound nSn > x, if suffices to demonstrate that there is an empty interval of some length x/n which contains none of the points. Make a list of some candidate intervals and then use Cheby- shev's inequality to show one of them is empty with probability going to 1.