Q. 9.4 - A First Course in Probability

Question

A certain town’s weather is classified each day as being rainy, sunny, or overcast, but dry. If it is rainy one day, then it is equally likely to be either sunny or overcast the following day. If it is not rainy, then there is one chance in three that the weather will persist in whatever state it is in for another day, and if it does change, then it is equally likely to become either of the other two states. In the long run, what proportion of days are sunny? What proportion are rainy?

Step-by-Step Solution

Answer

In the long run $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$proportion of days are rainy and $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$proportion of days are sunny.

Step 1: Given Information

We need to find that town's what proportion are rainy and what proportion of days are sunny.

Step 2: Simplify

The Markov chain has three states. Let state $1$ be rainy day, state $2$ be sunny day and state $3$ overcast. We have that the transition matrix is

$P=\left[\begin{array}{ccc}0& 1/2& 1/2\\ 1/3& 1/3& 1/3\\ 1/3& 1/3& 1/3\end{array}\right]$

Finding the stationary distribution. Solving$\mathrm{\pi }-\mathrm{\pi P}$ which is

$$\begin{gathered} {\mathrm{\pi }}_1= +\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{${\mathrm{\pi }}_2$}\right. +{\mathrm{\pi }}_3 \\ {\mathrm{\pi }}_2=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2{\mathrm{\pi }}_1 $}\right.+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\mathrm{\pi }}_2 +\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\mathrm{\pi }}_3 \\ {\mathrm{\pi }}_3=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2{\mathrm{\pi }}_1$}\right. +\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\mathrm{\pi }}_2 +{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3\mathrm{\pi }$}\right.}_3 \end{gathered}$$

Considering that it has to be ${\mathrm{\pi }}_1+{\mathrm{\pi }}_2+{\mathrm{\pi }}_3=1,$

${\mathrm{\pi }}_1=\frac{1}{4},{\mathrm{\pi }}_2={\mathrm{\pi }}_3=\frac{3}{8}$

which means$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$of days are rainy and$\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$8$}\right.$of days are sunny.