A car rental agency has rental and return facilities at both Dallas and Austin Airports. Assume that a car rented at either airport must be returned to one or the other airport. If a car is rented at Austin, the probability that it will be returned there is .80; If a car is rented at Dallas, the probability that it will be returned there is .70. Assume that the company rents all 100 cars each day and each car is only returned once each day. If we start with 50 cars at each airport: Create the transition matrix. 2. what is the expected distribution the next day? after 2 days?
transition matrix:
.8 .2
.7 .3
so
[.5 .5 ]
x
.8 .2
.7 .3
=
[.75 .25) next day
after 2 days:
[.75 .25]
x
.8 .2
.7 .3
= [.775 .225]
To create the transition matrix, we need to determine the probabilities of each possible transition from one state to another. In this case, the states are "Dallas" and "Austin," representing the locations of the rented cars.
Let's label the rows as "Dallas" and "Austin" and the columns as "Dallas" and "Austin" respectively.
From Dallas:
- There is a 70% probability (0.70) that a car rented in Dallas will be returned to Dallas. So, the transition probability from Dallas to Dallas is 0.70.
- There is a 30% probability (1 - 0.70) that a car rented in Dallas will be returned to Austin. Therefore, the transition probability from Dallas to Austin is 0.30.
From Austin:
- There is an 80% probability (0.80) that a car rented in Austin will be returned to Austin. Hence, the transition probability from Austin to Austin is 0.80.
- There is a 20% probability (1 - 0.80) that a car rented in Austin will be returned to Dallas. So, the transition probability from Austin to Dallas is 0.20.
The transition matrix is:
| Dallas | Austin |
--------------------------------
Dallas | 0.70 | 0.30 |
--------------------------------
Austin | 0.20 | 0.80 |
--------------------------------
The expected distribution (number of cars) the next day can be calculated by multiplying the current distribution by the transition matrix:
Let E0 be the initial distribution with 50 cars at Dallas and 50 cars at Austin.
Day 1 distribution (E1) = E0 * Transition Matrix
Day 2 distribution (E2) = E1 * Transition Matrix
To calculate the expected distribution after 2 days, you would multiply the Day 1 distribution (E1) by the Transition Matrix.
Note: Since the initial distribution is not given explicitly for each location, it is assumed that there is an equal distribution of 50 cars at each airport initially.