. A large but sparsely populated county has two small hospitals, one at the south end of the county
and the other at the north end. The south hospital’s emergency room has 4 beds, whereas the north
hospital’s emergency room has only 3 beds. Let X denote the number of south beds occupied at a
particular time on a given day, and let Y denote the number of north beds occupied at the same time
on the same day. Suppose that these two rvs are independent, that the pmf of X puts probability
masses .1, .2, .3, .2, and .2 on the x values 0, 1, 2, 3, and 4, respectively, and that the pmf of
Y distributes probabilities .1, .3, .4, and .2 on the y values 0, 1, 2, and 3, respectively.
(a) Display the joint pmf of X and Y in a joint probability table.
(b) Compute P(X _ 1 and Y _ 1) by adding probabilities from the joint pmf, and verify that this
equals the product of P(X _ 1) and P(Y _ 1).
(c) Express the event that the total number of beds occupied at the two hospitals combined is at
most 1 in terms of X and Y, and then calculate this probability.
(d) What is the probability that at least one of the two hospitals has no beds occupied?