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Chapter 9: Q. 9.4 (page 413)

A certain town’s weather is classified each day as being rainy, sunny, or overcast, but dry. If it is rainy one day, then it is equally likely to be either sunny or overcast the following day. If it is not rainy, then there is one chance in three that the weather will persist in whatever state it is in for another day, and if it does change, then it is equally likely to become either of the other two states. In the long run, what proportion of days are sunny? What proportion are rainy?

Short Answer

Expert verified

In the long run 14proportion of days are rainy and 38proportion of days are sunny.

Step by step solution

01

Given Information

We need to find that town's what proportion are rainy and what proportion of days are sunny.

02

Simplify

The Markov chain has three states. Let state 1be rainy day, state 2be sunny day and state 3overcast. We have that the transition matrix is

P=01/21/21/31/31/31/31/31/3

Finding the stationary distribution. Solvingπ-πPwhich is

π1=+4π2+π3π2=12π1+13π2+13π3π3=12π1+13π2+13π3

Considering that it has to be π1+π2+π3=1,

π1=14,π2=π3=38

which meanslocalid="1648147438354" 14of days are rainy and38of days are sunny.

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Most popular questions from this chapter

Customers arrive at a certain retail establishment according to a Poisson process with rate λ per hour. Suppose that two customers arrive during the first hour. Find the probability that

(a) both arrived in the first 20 minutes;

(b) at least one arrived in the first 30 minutes.

Events occur according to a Poisson process with rate λ = 3 per hour. (a) What is the probability that no events occur between times 8 and 10 in the morning? (b) What is the expected value of the number of events that occur between times 8 and 10 in the morning? (c) What is the expected time of occurrence of the fifth event after 2 P.M.?

A pair of fair dice is rolled. Let

X=1ifthesumofthediceis60otherwise

and let Y equal the value of the first die. Compute (a) H(Y), (b) HY(X), and (c) H(X, Y).

A random variable can take on any of n possible values x1, ... , xn with respective probabilities p(xi), i = 1, ... , n. We shall attempt to determine the value of X by asking a series of questions, each of which can be answered “yes” or “no.” For instance, we may ask “Is X = x1?” or “Is X equal to either x1 or x2 or x3?” and so on. What can you say about the average number of such questions that you will need to ask to determine the value of X?

Prove that if X can take on any of n possible values with respective probabilities P1, ... ,Pn, then H(X) is maximized when Pi = 1/n, i = 1, ... , n. What is H(X) equal to in this case?
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