Exercise 1 (5 + 10 + 5 pta). Let X be a continuous random variable with density f(I) = JAe ke, when A > 0,1 20, elsewhere. (a) Compute P(2 t), P(X > * + f(X > t) in terms of s,t 2 0, 1 3 0). (c) Usually, the probability that John waits less than 5 min at the bus stop before it arrives is 1/4. Given that he has already been waiting 10 min, what is the probability that he wait at least 5 more minutes? Model the time that John waits for the bus as an exponential r.v. Exercise 2 (5 + 5 pts). When a certain basketball player takes his first shot in a game he succeeds with probability 1/2. If he misses his first shot, he loses confidence and his second shot will go in with probability 1/3. If he misses his first 2 shots then his third shot will go in with probability 1/4. His success probability goes down further to 1/5 after he misses his first 3 shots. If he misses his first 4 shots then the coach will remove him from the game. Assume that the player keeps shooting until he succeeds or he is removed from the game. Let X denote the number of shots he misses until his first success or until he is removed from the game. 1. Compute the probability that the player would be removed given that he missed his first shot. 2. Calculate the probability mass function of X. Exercise 3 (4+ 4+ 4 + 4 pts). We play a card game where we receive 13 cards at the beginning out of the deck of 52. We play 50 games one evening. For each of the following random variables identify the name and the parameters of the distribution. (a) The number of acea I get in the first game. (b) The number of games in which I receive at least one ace during the evening. (c) The number of games in which all my cards are from the same suit. (d) The number of spades I receive in the 5th game. Exercise 4 (5 + 10 pts). Let X and Y be independent Geo(p) random variables. 1. Find the probability mass function of V = min(X, Y). Hint: For & a non-negative integer, the event {V = k) can be decomposed using the events (X = Y = k), {X > k, Y = k) and {X = k, Y > k]. 2. Let if X Y Find the probability mass function of W. Hint: The case W = 1 is simple. Note that the two other cases are symmetric such that the associated probabilities are equal and you can then easily get the associated probabilities