2. Consider a birth and death process (X,, 0 0 and death rate 6 > 0. Denote by 5. = inf(t 2 0 : X, = 0) the extinction time. (a) i) Recall the definition of the (8, 6)-birth and death process as a population model and gives its diagram as a continuous-time Markov chain. ii) Using a property of this population model, show that P, (5. 0? ii) Argue that Os F(y + 1) - F(y) S F(1), Vy 21. iii) Show that F(y) = (B+6)y -+ #F(y + 1)+F(y - 1) , ify21 F(y) = 0 , ify = 0. iv) Let us set G(y) = F(y + 1) - F(y), Vy 2 0. Show that Bp"G(y) + k SP. = G(0), Vy 2 0 where p = e v) Consider 6 > B and let us admit that* F(1)