23. In a good weather year the number of storms is Poisson distributed with mean 1; in a bad year it is Poisson distributed with mean 3. Suppose that any year’s weather conditions depends on past years only through the previous year’s condition. Suppose that a good year is equally likely to be followed by either a good or a bad year, and that a bad year is twice as likely to be followed by a bad year as by a good year. Suppose that last year—call it year 0—was a good year.

(a) Find the expected total number of storms in the next two years (that is, in years 1 and 2).

(b) Find the probability there are no storms in year 3.

(c) Find the long-run average number of storms per year.