The example in my book for an Ehrenfest Markov chain is: A system of of two urns, A & B where there are 2n balls total in both urns. We are assuming that there are $i$ balls in urn A and $2n - i$ balls in urn B at any instant.
How do you find the stationary distribution of the Ehrenfest Markov Chain? I know that the definition of a stationary distribution is as follows:
$\pi P = \pi$
Where $\pi$ is an eigenvector of the transition probability matrix whose associated eigenvalue is equal to 1. I also know that the TPM comes from the following:
$P_{ij} =\left\{ \begin{array}{ll} \frac{i}{2n} & if j=i-1 \\ 1-\frac{i}{2n} & if j=i+1 \\ 0 & otherwise\\ \end{array} \right.$
I have also seen that the stationary distribution is $\binom{2n}{i} 2^{-2n}$
Would someone be able to help come to this conclusion that the stationary distribution is above?
Thank you!