Question

Let X₀, X₁, X₂, ... be a Markov chain with state space S = {1, 2, 3} transition probabilities

(0 1/2 1/2)
P = (4/5 0 1/5)
(1/3 1/3 1/3)

and initial distribution αᵀ = (1/2, 0, 1/2). Find:

a. the conditional distribution of X₂, given X₁ = 3;