# Thread: Rainy Day Markov Chain

1. ## Rainy Day Markov Chain

Suppose the probability of rain today is .6 if rain fell yesterday, but only .2 if it did not.

What is the average duration (number of days) of a rainy period?

I have found the transition matrix as

P = [.8 .2
.4 .6]

And I have solved the other two parts of the problem which are:

X = 1, as a rainy day, X = 0 as a non rain day

Give rain fell today, what is the probability of ran on the day after tomorrow?

P(X_2=1 | X_0 = 1) = 0.44

Found this by squaring the matrix and taking the second entry on the column and row.

The other question answered was the fraction of days in which rain falls?

I found the fractions as non-rainy days -> 2/3 and rainy days -> 1/3, by solving the steady state equations.

All that as background. I'm just not sure how to find the average duration of a rainy period.

2. you find the mean time between states for a non rainy day which is 1/fraction as non-rainy days = $\frac{3}{2}$. Because if it takes this long from non rainy to get back to non rainy then inbetween it must be rainy.

I think thats it but you maybe have to check that