20.3: Another Markov Model Example

Last updated
Save as PDF

Page ID: 68026

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

from urllib.request import urlretrieve

urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

A sports broadcaster wishes to predict how many Michigan residents prefer University of Michigan teams and how many prefer Michigan State teams. She noticed that, year after year, most people stick with their preferred team; however, about 5% of Michigan fans switch to Michigan State, and about 3% of Michigan State fans switch to Michigan each year. However, there is no noticeable difference in the state’s population of 10 million’s preference at large; in other words, it seems Michigan sports fans have reached a stationary distribution. What might that be?

This problem is from Brilliant.org.

Do This

Try to draw a Markov chain for the above system of equations. Discuss your diagram with your classmate.

Question

Write a system of linear equations that represents how the populations change each year. Check your equations by writing the matrix P for the probability transitions matrix in your equations. Make sure your first row/column represents MSU and the second row/column represents UofM.

#Put your answer here

from answercheck import checkanswer

checkanswer.vector(P,'1d3f7cbebef4b610f3b0a2d97609c81f');

Question

Calculate the eigenvalues and eigenvectors of your \(P\) transition matrix.

#Put the answer to the above quesiton here.

Question

Assuming each team starts with 500,000 fans, what is the steady state of this model? (i.e. in the long term how many Spartan and Wolverine fans will there be?).

#Put your answer here

from answercheck import checkanswer

checkanswer.float(spartans,'06d263de629f4dbe51eafd524b69ddd9');

from answercheck import checkanswer

checkanswer.float(wolverines,'62d63699c8f7b886ec9b3cb651bba753');