Please help with the below question :

[Markov jump process] Let X = (X, : 1 2 0) be a time homogeneous Markov process with a finite state space S and a given intensity matrix O. The (x, y)-element qn 2 0 of @ defines the transition rate of moving from a state x to another state y, with both x, y e S. It is defined by lim 4 Px(1), for y # x 10 4 xy = (1) lim :(Pry(1) - 1), fory = x, where P. () denotes the transition probability of moving from x to y in t 2 0 periods of time, i.e., it is the (x, y)-component of transition matrix P(t). Note that qxx = - Eyex qxy. Consider a simple case of X with $ = (1, 2, 3) and the transition matrix P() given by PI(t) P12(t) PB3(t)) P(() = 0 P22(1) Pz(t) 0 0 where Pu(!) =e-0.51 P12(1) = =(e-0.1s _ e-0.51) P2(1) =e-0.1 (a) Explain the common properties and the differences between time homogeneous and time non-homogeneous Markov processes in terms of the transition probability. (b) Use the definition (1) to complete the intensity matrix Q of X: 0.2 Q = 0 . . . (c) Find the expected time X stays in states 1 and 2