Science:Math Exam Resources/Courses/MATH152/April 2012/Question 07 (c)

MATH152 April 2012

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q4 (d)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q6 (e)  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  Q7 (d)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q8 (d)  •  

   Other MATH152 Exams

  •  April 2016  •  April 2015  •  April 2014  •  April 2022  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2017  •  

Question 07 (c)
Consider a random walk that has transition matrix: $P={\begin{bmatrix}1/4&1/4&1/4\\3/4&3/4&1/4\\0&0&1/2\end{bmatrix}}$ Find the equilibrium probability vector (i.e. the vector at which probabilities remain unchanged from one step to the next).

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
The stationary probability density, ${\bar {\mathbf {p} }}$, satisfies the equation: ${\bar {\mathbf {p} }}=P{\bar {\mathbf {p} }}.$

Hint 2
Consider your work in part (b). Can you use eigenvectors to help you answer this question?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The stationary probability density, ${\bar {\mathbf {p} }}$, satisfies the equation: ${\bar {\mathbf {p} }}=P{\bar {\mathbf {p} }}.$ We notice that ${\bar {\mathbf {p} }}$ must be the eigenvector with corresponding eigenvalue $\lambda =1$. Thus, we need to solve for this eigenvector (making sure all the entries sum to 1 in order for it to be a probability distribution) using the equation $(P-I){\bar {\mathbf {p} }}=\mathbf {0}$ ${\begin{bmatrix}-3/4&1/4&1/4\\3/4&-1/4&1/4\\0&0&-1/2\end{bmatrix}}{\bar {\mathbf {p} }}=\mathbf {0} \quad \rightarrow \quad {\bar {\mathbf {p} }}={\begin{bmatrix}1/4\\3/4\\0\end{bmatrix}}.$ Therefore, the stationary probabilities are probability 1/4 of being in state 1, probability 3/4 of being in state 2, and zero probability of being in state 3.