1. Let X be a random variable which can take on countably many values. Show that X cannot be uniformly distributed.
2. Suppose we are attending a college which has 3000 students. We wish to choose a subset of size 100 from the student body. Let X represent the subset, chosen using the following possible strategies. For which strategies would it be appropriate to assign the uniform distribution to X? If it is appropriate, what probability should we assign to each outcome?
(a) Take the first 100 students who enter the cafeteria to eat lunch.
(b) Ask the Registrar to sort the students by their Social Security number, and then take the first 100 in the resulting list.
(c) Ask the Registrar for a set of cards, with each card containing the name of exactly one student, and with each student appearing on exactly one card. Throw the cards out of a third-story window, then walk outside and pick up the first 100 cards that you find.