Probability with Markov chains I need some hint about Markov chains. So here is my homework. 


  Let $\{ X_t : t = 0,1, 2, 3, \ldots, n\}? $ be a Markov chain. What is $P(X_0 =i\mid X_n=j)$? 


So I need to calculate if it's $j$ in the time of $n$ what is the probability of being $i$ at the start?

Is it :

Since if we start from state 0, then the probability to go to next state 1 is $1/2$, and to state 2 is $1/4$, etc.

$$ P_i = \left(\frac {1} {2}\right)^n \cdot \frac {1} {2}$$

Is that right ?
Hint:
$P(X_0=i\mid X_n=j)=\frac{P(X_n=j\mid X_0=i)\times P(X_0=i)}{P(X_n=j)}=p_{ij}^{(n)}\times\frac{\pi_i}{\pi_j} $

Note: The Markov chain must be stationary, i.e., the chain must be in steady state.
Use Bayes theorem and decomposition matrix or probability generate function to solve it:
$$ P(X_0=i∣X_n=j) = \frac{P_{ij}^n\times P(X_0=i)}{\sum P_{ij}^n\times P(X_0=i)}$$