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Current Score : – / 159 Due : Sunday, April 24 2016 11:59 PM EDT

1. –/1 pointsGoldFM10 8.2.005.

Determine whether or not the matrix is a regular stochastic matrix.

Yes

2. –/2 pointsGoldFM10 8.2.007.

Find the stable distribution for the given regular stochastic matrix.

< i > x < /i >

< i > y < /i >

3. –/1 pointsGoldFM10 8.2.012.

For a certain group of states, it was observed that 80% of the Democratic governors were succeeded by Democrats and 20% by Republicans. Also, 50% of the Republican governors were succeeded by Democrats and 50% by Republicans. In the long run, what proportion of the governors will be Democrats?

4. –/1 pointsGoldFM10 8.2.014.

A certain university has a computer room with 246 terminals. Each day there is a 2% chance that a given terminal will break and a 80% chance that a given broken terminal will be repaired. In the long run, about how many terminals in the room will be working?

HW 11 (Homework)

Peter Rubenstein MAT 183, section 100, Spring 2016 Instructor: Vincent Fatica

WebAssign

5. –/1 pointsGoldFM10 8.2.016.

Commuters can get into town by car or bus. Surveys have shown that for those taking their car on a particular day, 33% take their car the next day and 67% take a bus. Also, for those taking a bus on a particular day, 50% take their car the next day and 50% take a bus. In the long run what percentage of the people take a bus on a particular day? (Round your answer to 1 decimal place.)

6. –/14 pointsGoldFM10 8.2.019.

The Day-by-Day car rental agency only rents cars on a daily basis. Rented cars can be returned at the end of the day to any of the agency's three locations; A, B, or C. The figure below shows the percentages of cars returned to each of the locations based on where they were picked up. Assume that all the agency's cars are rented each day, and that initially 40% of the cars are at location A, 30% at location B, and 30% at location C.

(a) Set up the stochastic matrix that displays these transitions.

A B C

(b) Use the matrix from part (a) to estimate the percentage of the cars at location A after one day.

% of the cars

Estimate the percentage of the cars at location A after two days. % of the cars

7. –/1 pointsGoldFM10 8.2.020.

The day-to-day changes in weather for a certain part of the country forms a Markov process. Each day is sunny, cloudy, or rainy. If it is sunny one day, there is a 70% chance that it will be sunny the following day, a 20% chance it will be cloudy, and a 10% chance of rain. If it is cloudy one day, there is a 30% chance that it will be sunny the following day, a 50% chance it will be cloudy, and a 20% chance of rain. If it rains one day, there is a 60% chance that it will be sunny the following day, a 20% chance it will be cloudy, and a 20% chance of rain. In the long run, what is the daily likelihood of rain?

In the long run, the daily likelihood of rain is .

8. –/18 pointsGoldFM10 8.2.024.

Use a graphing calculator or spreadsheet to calculate the answers. The figure below describes the migration pattern of a species of bird from year to year among three habitats: I, II, and III.

(a) Set up the stochastic matrix that displays these transitions.

I II III

III

(b) If there are 1000 birds in each habitat at the beginning of a year, how many will be in each habitat at the end of the year? (Round your answers to the nearest whole number.)

habitat I

habitat II

habitat III

How many will be in each habitat at the end of the two years?

habitat I

habitat II

habitat III

habitat I

habitat II

habitat III

9. –/1 pointsGoldFM10 8.2.027.

With respect to a certain gene, geneticists classify individuals as dominant, recessive, or hybrid. In an experiment, individuals are crossed with hybrids, then their offspring are crossed with hybrids, and so on. For dominant individuals, 75% of their offspring will be dominant and 25% will be a hybrid. For the recessive individuals 75% of their offspring will be recessive and 25% hybrid. For hybrid individuals (to be crossed with hybrids) their offspring will be 25% dominant, 50% recessive, and 25% hybrid. In the long run what percent of the individuals in a generation will be dominant?

10.–/1 pointsGoldFM10 8.3.002.

Determine whether the transition diagram corresponds to an absorbing stochastic matrix.

No, the state A is absorbing, but it's not possible to get to A from B or D.

Yes, the states A, B and C are absorbing, and it's possible to get to any of them from D.

No, the state A is absorbing, but it's not possible to get to A from C or D.

Yes, the states A and B are absorbing, and it's possible to get to A and B from states C and D.

11.–/1 pointsGoldFM10 8.3.004.

Determine whether the transition diagram corresponds to an absorbing stochastic matrix.

No, the state A is absorbing, but it's not possible to get to A from B or D.

Yes, the states A, B and C are absorbing, and it's possible to get to any of them from D.

Yes, the states A and B are absorbing, and it's possible to get to A and B from states C and D.

No, the state A is absorbing, but it's not possible to get to A from C or D.

12.–/1 pointsGoldFM10 8.3.007.

Determine whether the given matrix is an absorbing stochastic matrix. matrix(3,3[1,0,0.4,0,0.6,0.4,0,0.4,0.2])

Yes

13.–/13 pointsGoldFM10 8.3.013.

This matrix below is an absorbing stochastic matrix. Identify R and S and compute the fundamental matrix and the stable matrix. matrix(3,3[1,0,0.1,0,1,0.4,0,0,0.5])

R =

S =

F =

stable matrix =

14.–/28 pointsGoldFM10 8.3.015.

The matrix below is an absorbing stochastic matrix. Identify R and S and compute the fundamental matrix and the stable matrix. matrix(4,4[1,0,0.2,0,0,1,0.4,0.2,0,0,0.3,0.6,0,0,0.1,0.2])

R =

S =

F =

stable matrix =

15.–/20 pointsGoldFM10 8.3.021.

The lawyers at a law firm are either associates or partners. At the end of each year, 30% of the associates leave the firm, 20% are promoted to partner, and 50% remain associates. Also, 10% of the partners leave the firm at the end of each year. Assume that a lawyer who leaves the firm does not return.

(a) Draw the transition diagram for this Markov process. Label the states A, P, and L.

(b) Set up an absorbing stochastic matrix for the Markov process. L A P

(d) What is the expected number of years an associate will be in the firm before leaving? yrs

16.–/21 pointsGoldFM10 8.3.022.

Colleges have been rapidly making broadband Internet service available in their residence halls. Of the colleges that offer no broadband Internet service, each year 10% introduce DSL Internet service, 30% introduce cable Internet service, and 60% continue to offer no broadband Internet service. Once a type of broadband Internet service is established, the type of service is never changed.

(a) Draw the transition diagram for the Markov process.

(b) Set up an absorbing stochastic matrix for the Markov process. DSL Cable None

DSL

Cable

None

DSL

Cable

None

(d) In the long run, what percent of colleges will provide cable Internet service? (Round to the nearest percent.)

(e) What is the expected number of years required for a college to set up a broadband service if it currently does not provide Internet service? (Round your answer to one decimal place.)

yrs

17.–/34 pointsGoldFM10 8.3.025.

Suppose that the following data were obtained from the records of a certain two-year college. Of those who were freshmen (F) during a particular year, 70% became sophomores (S) the next year and 30% dropped out (D). Of those who were sophomores during a particular year, 80% graduated (G) by the next year and 20% dropped out.

(a) Set up the absorbing stochastic matrix with states D, G, F, S that describes this transition.

D G F S

(b) Find the stable matrix.

(d) Determine the expected number of years a student entering as a freshman will attend the college before either dropping out or graduating.