A fair die is rolled repeatedly and ask for the following transition matrix.
(1) the largest number $X_n$ shown up to the nth roll.
For this question, I know the answer is $$P_{ij}=\frac{i}{6}, j=i \text{ and } P_{ij}=\frac{1}{6}, j>i$$
But how to get it? Why $j=i$ the probability is $i/6$?
(2) At the time $n$, the time $T_n$ since the most recent six. How to get $P_{ij}$?
Is it a Markov chain? How to understand it?
(3) At the time $n$, the time $T_n$ until the next six. How to get $P_{ij}$?
I know that if $j>i$ then $P_{ij}=(5/6)^{j-1}(1/6)$ which is a geometry process. But how the other cases?
Guide:
(1) You have already obtain $i$ as the largest number. To stay at the same state, you can get any of the $i$ smallest value.
(2) Suppose at time $n$, you are at state $i$. It can either go to state $i+1$ or go to state $0$. To go to state $0$, you toss a $6$.
(3) The time until the next $6$ should be decreasing by $1$ unit with probability $1$ if your current state it not $0$. If your current state is $0$, then the transition probabiity follows geometric distribution.