Question

Stochastic probability, Renewal theory: The weather in a certain locale consists of alternating wet and dry spells. Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7. Assume that the successive durations of rainy and dry spells are independent. What is the long-run fraction of time that it rains?

Answer 1

Answer:

Given that:

Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7.

In the long run, fraction of time that it rains = E[number of days in rainy spell] / ( E[number of days in rainy spell]+E[number of days in dry spell])

E[number of days in rainy spell] = 2

E[number of days in dry spell] = 7

In the long run, fraction of time that it rains = 2/(2+7)

= 2/9