The manager of Capitol Park Plaza, an apartment complex in Washington, DC, collects the rent from each tenant on the first day of every month. Past records indicate that the mean number of tenants who do not pay the rent on time in any given month is 4.7. Consider the rent collection for the next month.
a. Find the probability that every tenant will pay the rent on time.
b. Find the probability that at least seven tenants will be late with their rent.
c. Suppose the number of delinquent rent payments in a month is independent of the number in every other month. What is the probability that at most three tenants will be late with their rent in two consecutive months?
The mean number of commercially prepared meals per week for a typical American is 4.39 This might seem a little high. But consider young professionals who eat lunch out several times each week, and families that order out for pizza or stop at a fast-food restaurant routinely.
a. What is the probability that a randomly selected American does not eat out during a week? Eats out once during the week? Twice during the week?
b. Suppose two Americans are selected at random. What is the probability that the total number of meals for the two Americans during a week is 0? 1? 2?
c. Suppose the mean number of times an American eats out during a two-week period is 8. What is the probability that a randomly selected American does not eat out during a two-week period? Once during the two-week period? Twice during the two-week period?
d. How do your answers in parts (b) and (c) compare? What property does this suggest about a Poisson random variable?