Alvin shops for probability books for K hours, where K is a random variable that is equally likely to be 1, 2, 3, .., 5. The number of books N that he buys is random and depends on how long he shops according to the conditional PMF

p _N|K (n|k)=1/k, for n = 1, . . . , k.

Find the joint PMF of K and N:
p _N,K (n=4,k=7)=

p _N,K (n=5,k=8)=

Find the marjinal PMF of N:
p _N (n=1)=

p _N (n=2)=

Find the conditional PMF of K given that N=4.
p _K|N (k=2|4)=

p _K|N (k=6|4)=

Find the conditional mean and variance of K, given that he bought at least 2 but no more than 3 books.
Conditional mean:

Conditional variance:

The cost of each book is a random variable with mean $23. What is the expected value of his total expenditure?