# Markov chains - expected hitting times

Consider the following markov chain with state space $$\{1,2,3,4\}$$ and transition matrix

$$\begin{pmatrix}1/3&&1/3&&0&&1/3 \\ 1/4&&1/4&&1/4&&1/4 \\ 0&&0&&1/2&&1/2 \\ 0&&0&&0&&1\end{pmatrix}$$

and initial distribution $$\lambda=(1/2,1/2,0,0)$$.

Let $$T=\inf\{n\ge0:X_n \in \{1,4\}\}$$.

And let $$k_i:=\mathbb E_i[T]$$ be the expected hitting time wrt $$\mathbb P_i$$.

How can I calculate the variance of $$T$$ wrt $$\mathbb P_3$$?

I know that $$Var(T)=\mathbb E_{\mathbb P_3}[T^2]-(\mathbb E_{\mathbb P_3}[T])^2$$

How can I determine the first summand $$\mathbb E_{\mathbb P_3}[T^2]$$?

• What does "with respect to $\operatorname{P}_3$" mean? $\operatorname{P}(T = k \mid X_0 = 3) = 2^{-k} \, [k > 0]$, you don't need to know the initial distribution for that. – Maxim Jun 26 at 17:31