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 souths hospital’s emergency room has four beds, whereas the north hospital’s emergency room has only three 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 rv’s are independent; that the pmf of X puts probability masses 0.1, 0.2, 0.3, 0.2, and 0.2 on the x values 0, 1, 2, 3, and 4 respectively; and that the pmf of Y distributes probabilities 0.1, 0.3, 0.4, and 0.2 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?
