Q. 9.8 - A First Course in Probability

Question

On any given day, Buffy is either cheerful (c), so-so (s), or gloomy (g). If she is cheerful today, then she will be c, s, or g tomorrow with respective probabilities .7, .2, and .1. If she is so-so today, then she will be c, s, or g tomorrow with respective probabilities .4, .3, and .3. If she is gloomy today, then Buffy will be c, s, or g tomorrow with probabilities .2, .4, and .4. What proportion of time is Buffy cheerful?

Step-by-Step Solution

Answer

Buffy is cheerful $\frac{30}{59}$portion of the time.

Step 1: Given Information

We need to find the proportion of time for Buffy cheerful.

Step 2: Simplify

The transition matrix of this Markov chain is given in the text and given as

$P=\left[\begin{array}{lll}0.7& 0.2& 0.1\\ 0.4& 0.3& 0.3\\ 0.2& 0.4& 0.4\end{array}\right]$

Finding stationary distribution. Solving the system$\pi =\pi P$which implies these equalities

$\begin{array}{llll}{\pi }_c& =0.7{\pi }_c& +0.4{\pi }_s& +0.2{\pi }_g\\ {\pi }_s& =0.2{\pi }_c& +0.3{\pi }_s& +0.4{\pi }_g\\ {\pi }_g& =0.1{\pi }_c& +0.3{\pi }_s& +0.4{\pi }_g\end{array}$

Solving this system and using${\pi }_c+{\pi }_s+{\pi }_g=1$ we get

$\pi =\left[\begin{array}{l}30/59\\ 16/59\\ 13/59\end{array}\right]$

so,$\frac{30}{59}$of time Buffy is cheerful.