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?

Short answer

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

Step-by-step solution

Given Information

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

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.