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\begin{document}

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\centerline{Math 36 --- Daily Homework}

\centerline{Assigned: 10/15} 

\centerline{Due: 10/18}
  

\begin{questions}

\question 

Draw examples of a Markov chain that 

\begin{enumerate}[label = (\alph*)]
\item is regular but $P$ and $P^2$ have zero entries
\item is ergodic but not regular
\item has period 2, but no cycles of length 2
\item is an absorbing Markov chain with more absorbing states than transient states
\item has an absorbing state $j$ where the probability of ending in that in $j$ is the same for every starting transient state $i$, i.e. $B_{ij}$ is the same for all $i$. 
\end{enumerate}

For each, draw the ``lily pad diagram'' of states and arrows with the transition probabilities and  explain why your Markov chain has the desired property. Don't just give the transition matrix.


\end{questions}


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