Explanation Given: The next day after a fair day would be fair with a probability of 0.6 and cloudy with a probability of 0.25. The next day after a rainy day would be cloudy with a probability of 0.4 and rainy with a probability of 0.25. The next day after a cloudy day would be cloudy with a probability of 0.35 and rainy with a probability of 0.25. Explanation: The probabilities for the weather of the next day can be represented by a transition matrix. Three classifications of weather would imply a 3 × 3 transition matrix. Define the following variables: F : fair weather C : cloudy weather R : rainy weather Use the provided information to set up the transition matrix P . F C R P = F C R [ 0.6 0.25 a 13 a 21 0.35 0.25 a 31 0.4 0.25 ] For a transition matrix, the sum of all entries for each row needs to be 1. Use this information to obtain the remaining values as follows: 0.6 + 0.25 + a 13 = 1 a 13 = 1 − 0.6 − 0.25 = 1 − 0.85 = 0.15 a 21 + 0.35 + 0.25 = 1 a 21 = 1 − 0.35 − 0.25 = 1 − 0.6 = 0.4 a 31 + 0.4 + 0.25 = 1 a 31 = 1 − 0.4 − 0.25 = 1 − 0.65 = 0.35 This gives the following transition matrix: F C R P = F C R [ 0.6 0.25 0.15 0.4 0.35 0.25 0.35 0.4 0.25 ] The long-range probability for the weather of the next day can be computed as the equilibrium vector for the transition matrix P . A vector v is said to be an equilibrium vector for a transition matrix P if the following condition holds: v P = v Let the vector v be defined as follows, so that the multiplication holds: [ v 1 v 2 v 3 ] Then, v P = v [ v 1 v 2 v 3 ] [ 0.6 0.25 0.15 0.4 0.35 0.25 0.35 0.4 0.25 ] = [ v 1 v 2 v 3 ] Simplify the left-hand side to obtain [ 0.6 v 1 + 0.4 v 2 + 0.35 v 3 0.25 v 1 + 0.35 v 2 + 0.4 v 3 0.15 v 1 + 0.25 v 2 + 0.25 v 3 ] = [ v 1 v 2 v 3 ] This leads to the following system of equations: 0.6 v 1 + 0.4 v 2 + 0.35 v 3 = v 1 0.25 v 1 + 0.35 v 2 + 0.4 v 3 = v 2 0.15 v 1 + 0.25 v 2 + 0.25 v 3 = v 3 This can be simplified to − 0.4 v 1 + 0.4 v 2 + 0.35 v 3 = 0 0.25 v 1 − 0.65 v 2 + 0.4 v 3 = 0 0.15 v 1 + 0.25 v 2 − 0.75 v 3 = 0 The system involving these four equations can be solved with the help of the Gauss–Jordan method. The augmented matrix for the system is [ − 0.4 0

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Videos

Question

Chapter 10.2, Problem 40E

To determine

The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.2, Problem 39E

Chapter 10.2, Problem 41E

Students have asked these similar questions

Can you tell the answer

ce- 216 Answer the following, using the figures and tables from the age versus bone loss data in 2010 Questions 2 and 12: a. For what ages is it reasonable to use the regression line to predict bone loss? b. Interpret the slope in the context of this wolf X problem. y min ball bas oft c. Using the data from the study, can you say that age causes bone loss? srls to sqota bri vo X 1931s aqsini-Y ST.0 0 Isups Iq nsalst ever tom vam noboslios tsb a ti segood insvla villemari aixs-Yedi

120 110 110 100 90 80 Total Score Scatterplot of Total Score vs. Putts grit bas 70- 20 25 30 35 40 45 50 Putts

Chapter 10 Solutions

Finite Mathematics (11th Edition)

Ch. 10.1 - Decide whether each matrix could be a...Ch. 10.1 - Decide whether each matrix could be a probability...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Decide whether each matrix could be a probability...Ch. 10.1 - Decide whether each matrix could be a...Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Decide whether each matrix could be a transition...Ch. 10.1 - Decide whether each matrix could be a...

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - In Exercises and 16, write each transition diagram...Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Find the first three powers of each transition...Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Insurance An insurance company classifies its...Ch. 10.1 - Insurance The difficulty with the mathematical...Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - 32. Land Use In one state, a Board of Realtors...Ch. 10.1 - Business The change in the size of businesses in a...Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Housing Patterns In a survey investigating changes...Ch. 10.1 - Migration A study found that the way people living...Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.2 - Which of the following transition matrices are...Ch. 10.2 - Which of the following transition matrices are...Ch. 10.2 - Which of the following transition matrices are...Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Find the equilibrium vector for each transition...Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Find the equilibrium vector for each transition...Ch. 10.2 - Prob. 16E Ch. 10.2 - Find the equilibrium vector for each...Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Business and Economics Quality Control The...Ch. 10.2 - 26. Quality Control Suppose improvements are made...Ch. 10.2 - (a) Dry Cleaning Using the initial probability...Ch. 10.2 - Mortgage Refinancing In 2009, many homeowners...Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - Prob. 32E Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - Migration As we saw in the last section, a study...Ch. 10.2 - 36. Criminology A study male criminals in...Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Prob. 41E Ch. 10.2 - 42. Language One of Markov's own applications...Ch. 10.2 - Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.3 - Find all absorbing states for each transition...Ch. 10.3 - Find all absorbing states for each transition...Ch. 10.3 - Find all absorbing states for each transition...Ch. 10.3 - Find all absorbing states for each transition...Ch. 10.3 - Find all absorbing states for each transition...Ch. 10.3 - Find all absorbing states for each transition...Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Find the fundamental matrix F for the absorbing...Ch. 10.3 - Prob. 10E Ch. 10.3 - Find the fundamental matrix F for the absorbing...Ch. 10.3 - Find the fundamental matrix F for the absorbing...Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - (a) Write a transition matrix for a gambler's ruin...Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - 20. How can we calculate the expected total...Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Business and Economics 23. Solar Energy In...Ch. 10.3 - 24. Company Training Program A company with a...Ch. 10.3 - Contagion Under certain conditions, the...Ch. 10.3 - 26. Medical Prognosis A study using Markov chains...Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Gambler's Ruin (a) Write a transition matrix tor a...Ch. 10.3 - 32. Tennis Consider a game of tennis when each...Ch. 10.3 - Professional Football In Exercise 40 of the first....Ch. 10 - 1. If a teacher is currently ill, what is the...Ch. 10 - Prob. 2EA Ch. 10 - Prob. 3EA Ch. 10 - Prob. 4EA Ch. 10 - Prob. 5EA Ch. 10 - Prob. 6EA Ch. 10 - Prob. 7EA Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Prob. 12RE Ch. 10 - Prob. 13RE Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - Prob. 25RE Ch. 10 - In Exercises 23-26, use the transition matrix P,...Ch. 10 - Prob. 27RE Ch. 10 - Prob. 28RE Ch. 10 - Prob. 29RE Ch. 10 - Decide whether each transition matrix is regular....Ch. 10 - Prob. 31RE Ch. 10 - Prob. 32RE Ch. 10 - Prob. 33RE Ch. 10 - Prob. 34RE Ch. 10 - Prob. 35RE Ch. 10 - Find all absorbing states for each matrix. Which...Ch. 10 - Prob. 37RE Ch. 10 - Prob. 38RE Ch. 10 - Prob. 39RE Ch. 10 - Prob. 40RE Ch. 10 - Prob. 41RE Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE Ch. 10 - Prob. 44RE Ch. 10 - Prob. 45RE Ch. 10 - Prob. 46RE Ch. 10 - Prob. 47RE Ch. 10 - Prob. 48RE Ch. 10 - Life Sciences 49. Medical Prognosis A study...Ch. 10 - Prob. 50RE Ch. 10 - Prob. 51RE Ch. 10 - Prob. 52RE Ch. 10 - Prob. 53RE Ch. 10 - Prob. 54RE Ch. 10 - Prob. 55RE Ch. 10 - Prob. 56RE Ch. 10 - Prob. 57RE Ch. 10 - Prob. 58RE Ch. 10 - Prob. 59RE Ch. 10 - Prob. 60RE Ch. 10 - Prob. 61RE Ch. 10 - Prob. 62RE Ch. 10 - Prob. 63RE Ch. 10 - Prob. 64RE Ch. 10 - Prob. 65RE Ch. 10 - Prob. 66RE Ch. 10 - Prob. 67RE Ch. 10 - Prob. 68RE Ch. 10 - 69. Gambling Suppose a casino offers a gambling...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

Solution Summary: The author explains how three classifications of weather would imply a 3times 3 transition matrix.

Concept explainers

Videos

The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

Want to see the full answer?

Chapter 10 Solutions