We consider the Markov chain $\left(X_{n}\right)_{n\geq0}$ on $\left\{ 1,2,3,4,5\right\}$ with transition matrix
$P=\left(\begin{array}{ccccc} 1/2 & 1/2 & 0 & 0 & 0\\ 1/2 & 1/2 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1/2 & 1/2 & 0\\ 1/2 & 0 & 0 & 0 & 1/2 \end{array}\right)$
Set $\tau:=\textrm{inf}\left\{ n\geq1\left|X_{n}=1\right.\right\} $ . Compute $\mathbb{P}_{5}\left[\tau=n\right]$ and $\mathbb{E}_{5}\left(\tau\right)$ .
EDIT: We have $\mathbb{P}_{5}\left[\tau=1\right]=\mathbb{P}_{5}\left[X_{1}=1\right]=1/2$ so by induction we get $\mathbb{P}_{5}\left[\tau=n\right]=\mathbb{P}_{5}\left[X_{n}=1,X_i \neq1 \forall i=1,...,n-1\right]=(1/2)^n$. But how to compute $\mathbb{E}_{5}\left(\tau\right)$?
I'm a freshmen of this subject. Help me. Thanks a lot.
$$\mathbb{P}_{5}\left[\tau=1\right]=\mathbb{P}_{5}\left[X_{1}=1\right]=1/2$$
$$\mathbb{P}_{5}\left[\tau=n\right]=\mathbb{P}_{5}\left[X_{n}=1,X_{i}\neq1\forall i=1,...,n-1\right]=(1/2)^{n}$$
Since $$\sum_{n=1}^{\infty}nx^{n}=\frac{x}{\left(1-x\right)^{2}}$$ we get that
$$\mathbb{E}_{5}\left(\tau\right)={\displaystyle \sum_{i=1}^{+\infty}i}\mathbb{P}_{5}\left[\tau=i\right]={\displaystyle \sum_{i=1}^{+\infty}i}\left(\dfrac{1}{2}\right)^{i}=\dfrac{1/2}{\left(1-1/2\right)^{2}}=2$$