1. We throw a coin until a head turns up for the second time, where p it the probability that a throw results in a head and we assume that the outcome of each throw is independent of the previous outcomes. Let X be the number of tunes we have thrown the coin. (a) Determine P[X : 2],P(X : 3), and P(X : 4). (b) Determine P[X : n] for n 2 2. 2. In a Bernoulli process with success probability p, we will wait until we have observed In successes. Starting immediately after the mth success, we will continue to watch until we have seen n failures. What is the mean (expected) value for the TOTAL number of trials we will observe? 3. Suppose that the number of typos in our 2027 textbook has a Poisson distribution with pa- rameter A : 1/2. What is the probability that there is at least one typo in this textbook ? 4. Assume that a new iPhone5 will be defective with probability 0.1. If a group of ten people buy 10 new iPhones (one each): (a) calculate the (exact) probability that at most one of them will be defective , (b) calculate the same probability using Poisson "approximation ." Calculate the exact numerical results for items (a) and (b) to compare how close they are. 5. Let the function h be given by [0, forx