Let O(t) be the occupation time for state 0 in the two-state continuous-time Markov chain. Find E[O(t)|X(0) = 1]. Consider the two-state continuous-time Markov chain. Starting in state 0, find...

Let O(t) be the occupation time for state 0 in the two-state continuous-time Markov chain. Find E[O(t)|X(0) = 1]. Consider the two-state continuous-time Markov chain. Starting in state 0, find Cov[X(s), X(t)]. Mr. Smith works on a temporary basis. The mean length of each job he gets is three months. If the amount of time he spends between jobs is exponentially distributed with mean 2, then at what rate does Mr. Smith get new jobs?