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)=h1, 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. 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 Xi, where Xi is the amount of money she spends on the ith book she purchases.
