Q. 9.9 - A First Course in Probability

Question

Suppose that whether it rains tomorrow depends on past weather conditions only through the past 2 days. Specifically, suppose that if it has rained yesterday and today, then it will rain tomorrow with probability .8; if it rained yesterday but not today, then it will rain tomorrow with probability .3; if it rained today but not yesterday, then it will rain tomorrow with probability .4; and if it has not rained either yesterday or today, then it will rain tomorrow with probability .2. What proportion of days does it rain? 

Step-by-Step Solution

Answer

The proportion of days that it rains is $\frac{2}{5}$.

Step 1: Given Information

We have to find the probability for the portion of days that it will rain.

Step 2: Simplify

This Markov chain has 4 states depending on the weather yesterday and today and these states are

$\left(R,R\right),\left(R,NR\right),\left(NR,R\right),\left(NR,NR\right)$

where for example says that it has not rained the day before, but it rains today. So, the transition matrix is given in the text and it is equal to

$P=\left[\begin{array}{cccc}0.8& 0.2& 0& 0\\ 0& 0& 0.3& 0.7\\ 0.4& 0.6& 0& 0\\ 0& 0& 0.2& 0.8\end{array}\right]$

Calculating for stationary distribution. Solving $\pi =\pi P$ which implies these set of equations

$\begin{array}{l}{\pi }_{R,R}=0.8{\pi }_{R,R}+0.4{\pi }_{NR,R}\\ {\pi }_{R,NR}=0.2{\pi }_{R,R}+0.6{\pi }_{NR,R}\\ {\pi }_{NR,R}=0.3{\pi }_{R,NR}+0.2{\pi }_{NR,NR}\\ {\pi }_{NR,NR}=0.7{\pi }_{R,NR}+0.8{\pi }_{NR,NR}\end{array}$

Solving this system of equations using ${\sum }_i{\pi }_i=1$,

$\pi =\left[\begin{array}{l}4/15\\ 2/15\\ 2/15\\ 7/15\end{array}\right]$

So, the proportion that it rains today is the sum of the first and the third row in $\mathrm{\pi }$as these states indicate that it rains today. Hence, the proportion is $\frac{2}{5}$.