(b) What is the probability that Carl will be waiting alone for more than 10 minutes? (c) What is the probability that Carl will have to wait more than 10 minutes until his party is complete? (d) What is the probability that the person who arrives second will have to wait more than 5 minutes for the third person to show up? Hint: (Alice's arrival time, Bob's arrival time) is Uniform over a square region, so the probability of it being in a specified subregion is proportional to the area of the subregion. 2. (BH 7.9) Let X and Y be i.i.d. Geom(p), and N = X + Y. (a) Find the joint PMF of X, Y, N. (b) Find the joint PMF of X and N. (c) Find the conditional PMF of X given N = n, and give a simple description in words of what the result says. 3. (BH 7.33) Alice walks into a post office with 2 clerks. Both clerks are in the midst of serving customers, but Alice is next in line. The clerk on the left takes an Expo()1) time to serve a customer, and the clerk on the right takes an Expo()2) time to serve a customer. Let T be the amount of time Alice has to wait until it is her turn. (a) Write down expressions for the mean and variance of T, in terms of double integrals (which you do not need to evaluate). (b) Find the distribution moon and variones of T without using calculus O Ei S X n 50