Assignment#3
HOMEWORK ASSIGNMENT#4, Due at 5:00 pm on Feb 7th
16.5-7 (2 pts)
16.5-8 (2 pts)
3) 2 pts
Consider two stocks. Stock 1 always sells for $10 or $20. If stock 1 is selling for $10 today, there is a 0.8 chance that it will sell for $10 tomorrow. If it is selling for $20 today, there is a 0.9 chance that it will sell for $20 tomorrow. Stock 2 always sells for $10 or $25. If stock 2 sells today for $10, there is a 0.9 chance that it will sell for $10 tomorrow. If it sells for $25 today, there is a 0.85 chance that it will sell for $25 tomorrow. Define states and construct Markov chains, with the (one-step) transition matrix for each stock. On the average, which stock will sell for a higher price? Find and interpret all mean first passage times.
4) 2 pts
Two types of squirrels – gray and black – have been seen in Pine Valley. At the beginning of each year, we determine which of the following is true:
There are only gray squirrels in Pine Valley.
There are only black squirrels in Pine Valley.
There are both gray and black squirrels in Pine Valley.
There are no squirrels in Pine Valley.
Over the course of many years, the following transition matrix has been estimated.
[pic]
a. During what fraction of years will gray squirrels be living in Pine Valley?
b. During what fraction of years will black squirrels be living in Pine Valley?
Solve by hand and then use IOR tutorial to verify your results.
5) 2 pts
The Gotham City Maternity Ward contains 2 beds. Admissions are made only at the beginning of the day. Each day, there is a 0.5 probability that a potential admission will arrive and a 0.5 probability that no one arrives to the ward. A patient can be admitted only if there is an open bed at the beginning of the day. Half of all patients are discharged after one day, and all patients that have stayed one day are discharged at the end of their second day. Define states and construct the (one-step) transition matrix of the Markov chain.
a. What is the fraction of days where all beds are utilized?
b. On the average, what percentages of the beds are utilized?
Solve by hand and then use IOR tutorial to verify your results.
6) 2 pts
The State College admissions office has modeled the path of a student through State College as a Markov chain:
[pic]
Each student’s state is observed at the beginning of each fall semester. For example, if a student is junior at the beginning of the current fall semester, there is an 80% chance that he will be a senior at the beginning of the next fall semester, a 15% chance that he will still be a junior, and a 5% chance that he will have to quit. (We assume that once a student quits, he never reenrolls.)
a. If a student enters State College as a freshman, how many years can he expect to spend as a student at State?
b. What is the probability that a freshman graduates?
7) Bonus, 2 pts
Customers buy cars from three auto companies. Given the company from which a customer last bought a car, the probability that she will buy her next car from each company is as follows:
[pic]
a. If someone currently owns a company 1 car, what is the probability that at least one of the next two cars she buys will be a company 1 car?
b. At present, it costs company 1 an average of $5,000 to produce a car, and the average price a customer pays for one is $8,000. Company 1 is considering instituting a five-year warranty. It estimates that this will increase the cost per car by $300, but a market research survey indicates that the probabilities will change as follows:
[pic]
Should company 1 institute the five-year warranty?
8) Bonus, 2 pts
Consider the Monopoly Markov chain from the lab.
a. What is the expected time spent in jail?
b. What is the expected recurrence time for paying the “luxury tax”? Assume that during the game, money is only made and lost from rents. Ignore the “Chance” and “Community Chest” cards that make you move, and building houses or hotels.
c. If you could own any one of the 8 monopolies for free, which one would you pick to guarantee maximum (expected) rent income per turn? (rents of different properties are given below)
|Property Name |Monopoly |Position |Price |Rent |
|Mediterranean Ave. |Purple |2 |60 |2 |
|Baltic Ave. |Purple |4 |60 |4 |
|Oriental Ave. |Light-Green |7 |100 |6 |
|Vermont Ave. |Light-Green |9 |100 |6 |
|Connecticut Ave. |Light-Green |10 |120 |8 |
|St. Charles Place |Violet |12 |140 |10 |
|States Ave. |Violet |14 |140 |10 |
|Virginia Ave. |Violet |15 |160 |12 |
|St. James Place |Orange |17 |180 |14 |
|Tennessee Ave. |Orange |19 |180 |14 |
|New York Ave. |Orange |20 |200 |16 |
|Kentucky Ave. |Red |22 |220 |18 |
|Indiana Ave. |Red |24 |220 |18 |
|Illinois Ave. |Red |25 |240 |20 |
|Atlantic Ave. |Yellow |27 |260 |22 |
|Ventnor Ave. |Yellow |28 |260 |22 |
|Marvin Gardens |Yellow |30 |280 |22 |
|Pacific Ave. |Dark-Green |32 |300 |26 |
|North Carolina Ave. |Dark-Green |33 |300 |26 |
|Pennsylvania Ave. |Dark-Green |35 |320 |28 |
|Park Place |Dark-Blue |38 |350 |35 |
|Boardwalk |Dark-Blue |40 |400 |50 |
d. Now, assume that you have to pay the price of every property in a monopoly to own that monopoly (Prices are given above). Using your calculations from part (c), calculate the break-even point (expected number of turns before starting to make profit) for each monopoly. Which monopoly reaches the break-even point fastest?
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