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Chapter 9: Q. 9.5 (page 412)

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

Short Answer

Expert verified

The probability that it will rain 3day from now is 12.

Step by step solution

01

Given Information  

We have given that there is 50鈥�50 chance of rain today.

We need to find the probability that it will rain three days from now.

02

Simplify

Given a transition matrix

P=0.70.30.30.7

and the beginning distribution ( the distribution of X0) is

x0=0.50.5

To find the distribution of X3we have

localid="1648052534806" role="math" x3=P3x0=0.70.30.30.73.0.50.5

Now,

P3=0.5320.4680.4680.532

That is

localid="1648051526652" x3=0.50.5

So, that means there is the probability of 12that it will rain on the third day.

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Most popular questions from this chapter

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.

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?

Compute the limiting probabilities for the model of Problem 9.4.

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This problem refers to Example 2f.

(a) Verify that the proposed value of 蟺j satisfies the necessary equations.

(b) For any given molecule, what do you think is the (limiting) probability that it is in urn 1?

(c) Do you think that the events that molecule j, j 脷 1, is in urn 1 at a very large time would be (in the limit) independent?

(d) Explain why the limiting probabilities are as given.
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