3 Probabilistically Buying Probability Books Chuck will go shopping for probability books for K hours. Here, K is a random variable and is equally likely to be 1, 2, or 3. The number of books / that he buys is random and depends on how long he shops. We are told that for n = 1, ..., k PN = n| K = k] = otherwise for some constant c. (a) Compute c. (b) Find the joint distribution of K and N. (c) Find the marginal distribution of N. (d) Find the conditional distribution of K given that N = 1. (e) We are now told that he bought at least 1 but no more than 2 books. Find the conditional mean and variance of K, given this piece of information. (f) The cost of each book is a random variable with mean 3. What is the expectation of his total expenditure? Hint: Condition on events N = 1, N =2, N =3 and use total expectation