Understanding the Probability of Storms in Good and Bad Weather Years

 Weather conditions can greatly impact an individual's daily life, and understanding the likelihood of certain weather patterns can be important in making plans and preparations. In this blog post, we will explore a Markov Chain model that takes into account the number of storms in good and bad weather years and the probability of transitioning between the two.

Problem:

Suppose that a year's weather conditions are dependent on the previous year's weather only. A good weather year has a Poisson distributed number of storms with a mean of 1, while a bad weather year has a mean of 3 storms. The likelihood of transitioning from a good year to a bad year, or vice versa, is determined by the conditions. A good year is equally likely to be followed by a good or bad year, and a bad year is twice as likely to follow another bad year. Given that the previous year (year 0) was a good year, we are asked to find:

(a) The expected number of storms in the next two years (years 1 and 2).

(b) The probability of having no storms in year 3.

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

Solution:

(a) To find the expected number of storms in the next two years, we need to calculate the expected number of storms in each year, taking into account the probability of transitioning between good and bad weather years. Let G and B denote good and bad weather years, respectively.

Given that the previous year was good, the expected number of storms in year 1 is 0.5 * 1 + 0.5 * 3 = 2 storms. The expected number of storms in year 2 depends on the weather in year 1. If year 1 was good, the expected number of storms in year 2 is 0.5 * 1 + 0.5 * 3 = 2 storms. If year 1 was bad, the expected number of storms in year 2 is 0.5 * 1 + 0.5 * 3 = 2 storms. Hence, the expected number of storms in the next two years is 2 + 2 = 4 storms.

(b) To find the probability of having no storms in year 3, we need to determine the probability of having good weather in year 2, and then use that information to find the probability of having no storms in year 3.

Given that year 1 was good, the probability of having good weather in year 2 is 0.5. Hence, the probability of having no storms in year 3 is e^-1 = 0.368.

(c) To find the long-run average number of storms per year, we need to determine the steady-state probabilities of being in a good or bad weather year, and then use those probabilities to find the average number of storms.

Let pG and pB be the steady-state probabilities of being in a good or bad weather year, respectively. We have:

pG = 0.5pG + 0.5pB

pB = 0.5pG + 1pB

Solving for pG and pB, we find that pG = 0.4 and pB = 0.6. Hence, the long-run average number of storms per year is 0.4 * 1 + 0.6 * 3 = 2.2 storms.