A large but sparsely populated county has two small hospitals, one at the south end of the county and one other at the north end. The south hospitals emergency room has four beds, whereas the north hospitals emergency room has only three beds. Let X denote the number of beds occupied in the south hospital at a particular time, and let Y denote the number of beds occupied in the north hospital at the same time. Suppose that these two rvs are independent. Further, suppose P[X = x] is equal to 0.1, 0.2,.3, 0.2 and 0.2 for x = 0, 1, 2, 3 and 4, respectively. P[Y = y] has values 0.1, 0.3, 0.4 and 0.2 for y = 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?