2. The Markov chain $\left\{X_{n} ; n \geq O ight\}$ has state- space $S=\{0,1,2, \ldots\)$, with $p_{i 0)=1 / 4$ and $p_{i, i+1)=3 / 4$ for all $i \geq 0$, so that the transition matrix is $$ P=\left(\begin{array}{ccccc} 1/4 & 3/ 4 & 0&& \ldots 1/4 & 0 & 3 / 4 & 0& \ldots 1/4 & 0& 0& 3 / 4 & \ldots \vdots & \vdots & \dots & \vdots & Iddots \end{array} ight) $$ Find the irreducible classes of intercommunicating states. For each class, state: (a) whether it is transient, positive recurrent or null recurrent (hint - think about the distribution of the return times - say to state 0 - in this case. From there, you can work out whether the states have a finite or infinite expected return time. Can you work out what sort of states you have here? The point of this question is that you have to do some calculations to understand what type of states you have here... don't just guess!); (b) its periodicity. SP. AS006