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\begin{center} {\large \textbf{Math 437 -- Homework 10}} \smallskip \\ 
Due 10:15am on Thursday, April 6, 2017  \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together in small groups, but please try the problems on your own first and write up 
your own solutions. \\ \medskip 


\begin{prob}
Consider two websites, each linking to each other.  Computing the PageRank with dampening factor 
$\alpha$ involves 
computing the steady state of the Markov Chain with transition matrix
\[ G  \ =  \ \alpha 
\begin{pmatrix} 0 & 1 \\ 1  & 0 \end{pmatrix} + (1-\alpha) \begin{pmatrix} 1/2 & 1/2 \\ 1/2  & 1/2
 \end{pmatrix} 
  \ =  \
\begin{pmatrix} (1-\alpha)/2 &  (1+\alpha)/2  \\  (1+\alpha)/2   &  (1-\alpha)/2  \end{pmatrix}.
\]
\begin{itemize}
\item[(a)] Compute the eigenvalues and eigenvectors of $G$ (as a functions of $\alpha$). 
\item[(b)] Find a formula for $G^n$. 
\item[(c)] If $\alpha=.9$, how large should $n$ be so that the first two digits of entries 
of $G^n$ agree with the limit $\begin{pmatrix} 1/2 &  1/2 \\  1/2  &  1/2 \end{pmatrix}.$   
What if $\alpha = .6$?
\end{itemize}
\end{prob}\ \\



\begin{prob} (GS 11.2, \#15)
Alice and Bob are playing tennis. 
Starting from \emph{deuce}, if a player wins the next point, that player has advantage. On the following point, the player with \emph{advantage} either wins the game or the game returns to deuce. Assume that for any point, Alice has probability $.6$ of winning the point and Bob has probability $.4$ of winning the point.
\begin{itemize}
\item[(a)] Set this up as a Markov chain with
the five states $\{ s_1 =$ advantage Alice, $s_2=$ deuce,
$s_3 =$ advantage Bob, $s_4 =$ Alice wins, $s_5 =$ Bob wins$\}$. 
\item[(b)] Starting from deuce, 
find the expected duration of the game and the probability that Bob will win.
\end{itemize}
\end{prob}\ \\



\begin{prob}
Have an approved topic for your final project.  
\end{prob}\ \\


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