Suppose that in any given period an unemployed person will find a job with probability 0.7 andwill therefore remain unemployed with a probability of 0.3. Additionally, persons who findthemselves employed in any given period may lose their job with a probability of 0.1 (and willhave a 0.9 probability of remaining employed). Write out the Markov system of differenceequations for this economy. Compute the stationary distributions.

Given information: The probabilities of employment and unemployment are given.

Answered: Suppose that in any given period an…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 12EQ: 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction...

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12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)
A system consists of five components, each can be operational or not. Each day one operational component is used and it will fail with probability 20%. Any time there no operational components at the end of a day, maintenance will be performed and all non-operational components will be repaired (with probability 1). The system does not perform any other tasks on the day of repairs. Model the system as a Markov chain Write down equations for determining long-run proportions. Suppose that you are interested in the average number of days that the system is under repair. Explain how you would find it using your model.
Answer the following questions.
The computer center at Rockbottom University has been experiencing computer downtime. Let us assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities. To From Running Down Running 0.80 0.20 Down 0.30 0.70 (a) If the system is initially running, what is the probability of the system being down in the next hour of operation? (b) What are the steady-state probabilities of the system being in the running state and in the down state? (Enter your probabilities as fractions.) Running?1=Down?2=
A factory worker will quit with probability 1⁄2 during her first month, with probability 1⁄4 during her second month and with probability 1/8 each month after that. Whenever someone quits, their replacement will start at the beginning of the next month and follow the same pattern. Model this position’s status as a Markov chain. What is the long-run probability of having a new employee on a given month? please provide steps and explanations for answers
For the following Markov models: a ito dove b) find the stationary probability distribution on paper, SUre 5A An ion channel can be in either open or closed state. If it is open, then it has probability 0.1 of closing in 1 microsecond; if closed, it has probability 0.3 of opening in 1 microsecond. 5B An individual can be either susceptible or infected, the probability of infection for a susceptible person is 0.05 per day, and the probability an infected person becoming susceptible is 0.12 per day. 5C The genotype of an organism can be either normal (wild type) or mutant. Each generation, a wild type individual has probability 0.03 of having a mutant offspring, and a mutant has probability 0.005 of having a wild type offspring.
Suppose that Phone A has a 20% market share initially, with Phone B having an 80% market share. The transition matrix to the right represents customers who currently own phones A and B and the probability that they will upgrade to a new phone A or B. Use this transition matrix to find the share for each phone after the following upgrades. A B A 0.8 0.2 B 0.25 0.75 (a) First upgrade (b) Second upgrade (c) Third upgrade (a) With the first upgrade, Phone A has % of the market and Phone B has % of the market. (Round your answers to two decimal places as needed.)
The weather can be considered a stochastic system, because it evolves in a probabilistic manner from one day to the next. Suppose for a certain location that this probabilistic evolution satisfies the following description: The probability of rain tomorrow is 0.6 if it is raining today. The probability of it being clear (no rain) tomorrow is 0.8 if it is clear today. (For simplicity, you can assume clear = 0 and rain = 1) a. If today’s weather is clear, simulate the weather condition for the next 7 days. b. What kind of probability is used in the table above (Poisson distribution, Normal Distribution, Uniform distribution, Exponential distribution)? Explain your answer.
A factory worker will quit with probability 1/2 during her first month, with probability 1/4 during her second month and with probability 1/8 after that. Whenever someone quits, their replacement will start at the beginning of the next month. Model the status of each position as a Markov chain with 3 states. Identify the states and transition matrix. Write down the system of equations determining the long-run proportions. Suppose there are 900 workers in the factory. Find the average number of the workers who have been there for more than 2 months.
Nick takes half-court shots on a basketball court. He is a streaky shooter, so his shot outcomes are not independent. If Nick made his last shot, then he makes his current one with probability a. If Nick missed his last shot, then he makes his current one with probability b, where b < a. Modeling Nick’s sequence of half-court shot outcomes as a Markov chain, what is the long-run probability that he makes a half-court shot?
A certain stock price has been observed to follow a pattern. If the stock price goes up one day, there's a 20% chance of it rising tomorrow, a 30% chance of it falling, and a 50% chance of it remaining the same. If the stock price falls one day, there's a 35% chance of it rising tomorrow, a 50% chance of it falling, and a 15% chance of it remaining the same. Finally, if the price is stable on one day, then it has a 50-50 change of rising or falling the next day. Find transition probability matrix for this Markov chain, if we list states in the order: (rising, falling, constant)
At Suburban Community College, 30% of all business majors switched to another major the next semester, while the remaining 70% continued as business majors. Of all non-business majors, 10% switched to a business major the following semester, while the rest did not. Set up these data as a Markov transition matrix. HINT [See Example 1.] (Let 1 business majors, and 2 = non-business majors.) = Calculate the probability that a business major will no longer be a business major in two semesters' time.

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