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Chapter 9: Q. 9.8 (page 412)

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

Expert verified

Buffy is cheerful 3059portion of the time.

Step by step solution

01

Given Information

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

02

Simplify

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

P=0.70.20.10.40.30.30.20.40.4

Finding stationary distribution. Solving the system蟺=蟺Pwhich implies these equalities

蟺c=0.7蟺c+0.4蟺s+0.2蟺g蟺s=0.2蟺c+0.3蟺s+0.4蟺g蟺g=0.1蟺c+0.3蟺s+0.4蟺g

Solving this system and using蟺c+蟺s+蟺g=1we get

蟺=30/5916/5913/59

so,3059of time Buffy is cheerful.

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