Alice is going shopping for statistics books for H hours, where H is a random variable, equally likely to be 1, 2 or 3. The number of books B she buys is random and depends on how long she is in the store for. We are told that P(B = b | H = h) = 1/h, for b = 1,...,h.
a) Find the joint distribution of B and H using the chain rule. b) Find the marginal distribution of B. c) Find the conditional distribution of H given that B = 1 (i.e., P(H = h | B = 1) for each possible h in 1,2,3). Use the definition of conditional probability and the results from previous parts. d) Suppose that we are told that Alice bought either 1 or 2 books. Find the expected number of hours she shopped conditioned on this event. Use the definition of conditional expectation and Bayes Theorem. e) The bookstore has a discounting policy that gives an extra 10% off the total purchase price if Alice buys two books and 20% off the total purchase price if she buys three books. Suppose that Alice's decision about what books to buy does not depend on their price and that, in an ironic twist, the bookstore owner also prices each statistics book randomly with a mean price of $40 per book. What is the expected amount of money Alice spends (assuming that book purchases are tax-free)? Warning: Be sure to use a formal derivation. Your work should involve the law of total expectation conditioning on the number of books bought, and make use of random variables X₁, where X, is the amount of money she spends on the ith book she purchases.