1. An intensity matrix of a continuous-time homogeneous Markov chain X is given by -4 ... Q 0 ... 1 3 ... 0 (a) Complete the matrix Q. Draw the transition diagram of X. (b) Find the average time X spends in each state and the probability of making a jump between states. (c) Let q₁ = q. Use the following integrated form of Kolmogorov backward equation qi to find the transition (probability) matrix [P;];; = P(X₁ = j|X₁ = i) of X, Pij(t) = Pij(0)e-gi² + Σ√ √ eqikPkj(t-u)du. k#i (d) Verify your solution for the transition probability pi;(t) by checking the limit (Pt-I) Q = lim to t where I is a (3 × 3)-identity matrix. (1)