2. Let {X t = 0, 1, 2,...} be a discrete-time Markov chain. Prove that given X₁ = i, Xn+1 is independent of {Xo, X₁,..., Xn-1}. In other words, prove that P(Xn+1 = j, Xo = io, X₁ = 1,..., Xn-1=in-1 Xn = i) =P(Xn+1 =jXn = i)P(Xo = io, X₁ = ₁,..., Xn-1=in-1|Xn = i) for any j, io, i1,..., in-1,i in the state space.

Please do the following question with handwritten working out

C)

A Markov chain model for a species has four states: State 0 (Lower Risk), State 1 (Vulnerable), State 2 (Threatened), and State 3 (Extinct). For t 2 0, you are given that: 01 zit = 0.03 12 t= 0.05 23 Hit = 0.06 This species is currently in state 0. Calculate the probability this species will be in state 2 ten years later. Assume that reentry is not possible. (Note: This question is similar to #46.2 but with constant forces of mortality) Possīble Answers A 0.02 0.03 0.04 D 0.05 E 0.06