Consider a Markov chain with state $\{0,1,2,3,4\}$ and a one$-$step transition probability matrix given by

a$.$ Assuming that the process starts in state $2,$ what is the probability that the state is absorbed into state $0$ and what is the probability that the state is absorbed into state $1?$

b$.$ What is the probability of the first transition time from state $2$ to state $4?$

c$.$ What is the expected recurrence time of state $2?$

d$.$ If the process starts with equal probability in states $0,1$ and $2,$ what is the probability that state $0$ is absorbed$?$

e$.$ Assuming the process starts in state $2,$ what is the expected duration of the process until it is absorbed$?$ ge$$i$$ olas$$l$$k matrisi $$u

zincirini d$$n$$n:

$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ \mathrm{.}2& \mathrm{.}2& \mathrm{.}2& \mathrm{.}2& \mathrm{.}2\\ \mathrm{.}25& 0& \mathrm{.}25& \mathrm{.}25& \mathrm{.}25\\ 0& \mathrm{.}25& \mathrm{.}25& \mathrm{.}25& \mathrm{.}25\end{array}\right]$