2 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 four beds whereas the north hospital s emer gency room has only three beds Let X denote the num ber 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 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 probabil ity 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 XS 1 and P Y 1 c Express the event that the total number of beds occu pied 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