Markov chain - stationary distribution -

In summary, the conversation discusses a problem involving a taxi that is either located at the airport or in the city. The next trip from the city has a 1/4 probability of going to the airport and 3/4 probability of going somewhere else in the city. From the airport, the next trip always goes to the city. The problem asks to find the stationary distribution and expected number of trips until the next visit to the airport. The transition matrix for the problem is given as 0 1 1/4 3/4, and the stationary distribution is found by solving two linear equations to be [1/5 4/5].
  • #1
Carolyn
37
0

Homework Statement



-A taxi is located either at the airport or in the city. From the city, the next trip is to the airport with 1/4 probability, or to somewhere else in the city with 3/4. From the airport, the next trip is always to the city.

(a) find the stationary distribution
(b) starting from the airport, what is the expected number of trips until its next visit to the airport?

Homework Equations





The Attempt at a Solution



The matrix I got is

0 1
1/4 3/4

but I am having trouble solving for the stationary distribution and expected number of trips. Somehow, I don't think the stationary distribution even exists. But it should. What am I missing? Thanks.
 
Physics news on Phys.org
  • #2
hello carolyn

your transition matrix is same as expected.And the stationary distribution exists. you can find the stationary distribution by solving it as two linear equations.
let 'Pij' be the element, i=1,2 and j=1,2 (for the given transition matrix).

P0=P0*P00+P1*P01; P1=P0*P10+P1*P11

here P0 and P1 are unknowns , the elements of stationary distribution.
Pij(P00,P01,P10,P11) are the transition matrix elements.

By substituting you will get P0=P0*0+P1*(1/4); P1=P0*1+P1*(3/4).

Also the condition to find P0 and P1 is P0+P1=1(the raw sum =1);

then u will get a stationary distribution of
[1/5 4/5]
 

1. What is a Markov chain?

A Markov chain is a mathematical model used to describe the transition of a system from one state to another over a series of discrete time steps. It is based on the assumption that the future state of the system only depends on its current state, and not on any previous states.

2. What is a stationary distribution in a Markov chain?

A stationary distribution in a Markov chain is a probability distribution that remains unchanged over time. It is often referred to as the "equilibrium" or "steady-state" distribution, as the system will eventually reach this distribution regardless of its initial state.

3. How is the stationary distribution calculated in a Markov chain?

The stationary distribution can be calculated by finding the eigenvector associated with the eigenvalue of 1 for the transition probability matrix of the Markov chain. This eigenvector represents the probabilities of being in each state in the long run.

4. Can a Markov chain have more than one stationary distribution?

Yes, it is possible for a Markov chain to have multiple stationary distributions. This can occur when there are multiple eigenvalues of 1 for the transition probability matrix, leading to multiple eigenvectors that represent different stationary distributions.

5. How is the concept of stationary distribution used in real-world applications?

The concept of stationary distribution is widely used in various fields such as economics, finance, biology, and physics. It can be used to model and analyze various systems, including stock prices, population growth, and chemical reactions. In these applications, the stationary distribution helps to predict the long-term behavior of the system and identify any potential patterns or trends.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
195
  • Precalculus Mathematics Homework Help
Replies
24
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
24
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
Back
Top