12.11 Consider a random walk with n + 1 states. The transition probability from state i to state j is Pij =

⎧

⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

1 − p if i = j = 0 p if j = i + 1, and i = 0,...,n − 1 p if i = j = n 1 − p if j = i − 1, and j = 1,...,n 0 otherwise.

Draw the states and transitions of the Markov chain. Classify its states and calculate its stationary distribution.