, no more than 20 MB in size. It can be typeset, written using a tablet, or scanned from handwritten work on paper, but must be clear, easily legible, and correctly rotated. Always remember to check your submission: download it after uploading it in Canvas, and make sure that it is the correct file and that it got uploaded successfully. No submissions on paper or by email will be accepted, and no extensions will be granted aside from the Monday extensions described in the syllabus. 1. (BH 7.2) Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day. (a) What is the probability that Carl arrives first? For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition on this fact. (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. O Eli S X n 50OF