Showing a certain random process is a Markov Process

Question

I have the following example of a random process: A person has two houses, house A and house B in which he can stay, we denote by $X_{i}\in\left\{ A,B\right\}$ the house he stayed in on the i-th day and it is given that:$$P\left(X_{i}=B\,|\, X_{i-1}=A\right)=q$$ $$P\left(X_{i}=A\,|\, X_{i-1}=A\right)=1-q$$ $$P\left(X_{i}=B\,|\, X_{i-1}=B\right)=1-p$$ $$P\left(X_{i}=A\,|X_{i-1}=B\right)=p$$ I need to show that this random process is a Markov-Process and find a stationary distribution for it. I know the definitions of a Markov-Process and a stationary distribution but I've never seen an example of how to show/find them so I'm pretty much completely lost.

Answer 1

To prove given process is markov , show that $P(X_i=x_i|X_{i-1}=x_{i-1},X_{i-2}=x_{i-2},\cdots,X_0=x_0)=P(X_i=x_i|X_{i-1}=x_{i-1})$

To show that start with $i=2$ and use induction.

EDIT: Transition probability matrix $P$ is $\begin{pmatrix}1-q \ \ \ \ q \\ \ \ \ \ \ \ p \ \ \ \ \ \ \ \ 1-p\end{pmatrix}$

Let stationary probabilities be $\pi =(p_1,p_2)$, then solve for $\pi $ using $\pi P=\pi$