Alvin shops for probability books for K hours, where K is a random variable that is equally likely to be 1, 2, 3, or 4. The number of books N that he buys is random and depends on how long he shops according to the conditional PMF
pN |K(n|k) = 1, for n = 1,...,k. k
(a) Find the joint PMF of K and N.
(b) Find the marginal PMF of N.
(c) Find the conditional PMF of K given that N = 2. 3
(d) Find the conditional mean and variance of K, given that he bought at least 2 but no more than 3 books.
(e) The cost of each book is a random variable with mean $30. What is the expected value of his total expenditure? Hint: Condition on the events {N = 1}, . . . , {N = 4}, and use the total expectation theorem.