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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 systemπ=πPwhich implies these equalities

πc=0.7πc+0.4πs+0.2πgπs=0.2πc+0.3πs+0.4πgπg=0.1πc+0.3πs+0.4πg

Solving this system and usingπc+πs+πg=1we get

π=30/5916/5913/59

so,3059of time Buffy is cheerful.

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Most popular questions from this chapter

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?

Show that for any discrete random variable Xand functionf

H(f(X))≤H(X)

Compute the limiting probabilities for the model of Problem 9.4.

Consider Example 2a. If there is a 50–50 chance of rain today, compute the probability that it will rain 3 days from now if α = .7 and β = .3.

Suppose that in Problem 9.2, Al is agile enough to escape from a single car, but if he encounters two or more cars while attempting to cross the road, then he is injured. What is the probability that he will be unhurt if it takes him s seconds to cross? Do this exercise for s = 5, 10, 20, 30.
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