An article in Clinical Infectious Diseases [“Strengthening the Supply of Routinely Administered Vaccines in the United States: Problems and Proposed Solutions” (2006, Vol.42(3), pp. S97–S103)] reported that recommended vaccines for infants and children were periodically unavailable or in short supply in the United States. Although the number of doses demanded each month is a discrete random variable, the large demands can be approximated with a continuous probability distribution. Suppose that the monthly demands for two of those vaccines, namely measles–mumps–rubella (MMR) and varicella (for chickenpox), are independently, normally distributed with means of 1.1 and 0.55 million doses and standard deviations of 0.3 and 0.1 million doses, respectively. Also suppose that the inventory levels at the beginning of a given month for MMR and varicella vaccines are 1.2 and 0.6 million doses, respectively.

(a) What is the probability that there is no shortage of either vaccine in a month without any vaccine production?

(b) To what should inventory levels be set so that the probability is 90% that there is no shortage of either vaccine in a month without production? Can there be more than one answer? Explain.