1. A death process with state space Z = {0, 1, 2, ...} has death rates μ 0 = 0, μ 1 = 2, and μ 2 = μ 3 = 1. Given X(0) = 3, determine p j (t) = P(X(t) = j) for j = 0, 1, 2, 3. 2. A linear death...

1. A death process with state space Z = {0, 1, 2, ...} has death rates

μ₀
= 0, μ₁
= 2, and μ₂
= μ₃
= 1.

Given X(0) = 3, determine p_j(t) = P(X(t) = j) for j = 0, 1, 2, 3.

2. A linear death process {X(t), t ≥ 0} has death rates

(1) Given X(0) = 2, determine the distribution function of the time to entering state 0 ('lifetime' of the process)

(2) Given X(0) = n, n > 1, determine the mean value of the time at which the process enters state 0.