Consider a birth and death process with states {0,1}. Let the birth and death rates beAo = a > 0, A₁ = 0, μo = 0, and µ₁ = b > 0.= ko =A1(a) Show that the infinitesimal generator matrix A =[a asatisfies the property A² = -2-(a + b)A.bThus, for n = 1, 2, 3,..., A = (a - b)-1A holds.n(b) Using the result from part (a) to compute the matrix exponential et in closed form and thus find thetransition probability matrix P(t) = e¹A, t≥ 0.(c) Compute lim →∞ P(t) and use it to find the stationary distribution.

Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao a > 0, A₁ = 0, μo = 0, and μ₁ => 0. (a) Show that the infinitesimal generator matrix A = Thus, for n = 1,2,3,..., A" = (-a - b)-1A holds. -a a b satisfies the property A² = (a + b)A. (b) Using the result from part (a) to compute the matrix exponential etA in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim∞ P(t) and use it to find the stationary distribution.