Markov Process Problem

Any help is greatly appreciated. I'm completely lost. 


We know that $P=P \Pi $. Also note that $P$ is a $1\times n$ matrix aka a row vector. So the k-th component of $P$ is $P$ times the k-th column of $\Pi$ :
$$ p_k =\sum _{i=1}^n p_i \pi _{ik} = \sum _{i=1}^n p_i \alpha = \alpha \sum _{i=1}^n p_i =\alpha 1=\alpha $$ since $P$ is a probability distribution and the sum of its components is one. 

The interpretation is that if the probability of reaching the state $x_k$ from any state is the constant $\alpha$, then in the stationery distribution, which is the same as the probability distribution in the long run, the probability of being in the state $x_k$ is also $\alpha$.

The answer is accepted.
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