Let {X(t),t ≥ 0} be a birth and death process whose state space is the set {0,1,2} and for which

We consider two independent copies, {X₁(t), t ≥ 0} and {X₂(t), t ≥ 0}, of this process, and we define

We can show that {Y(t),t ≥ 0} is also a birth and death process.

(a) Give the birth and death rates of the process {Y(t), t ≥ 0}.

(b) Calculate the expected value of the random variable Y{t) after two transitions if X₁(0) = X₂(0) = 0.

(c) Calculate the limiting probabilities of the process {Y(t), t ≥ 0}.