Suppose you have a two state continuous time markov chain (X_t = {1,2}). Jumps occur randomly with exponential distribution with parameter mu and lamda. I have the transition matrix (though it's more latex than I care to type at the moment). Fix a time T in the future. What is the distribution of time you spend in state 1 prior to time T? For simplicity, assume you start in state 1 with certainty. That is, I would like to know the distribution of the following random variable. I've looked all over for the answer to this question. And I have not found the solution anywhere.Incidentally, does anybody know how to find the distribution of time a Brownian motion spends in a given region (say 1-d BM in [a,b]) in a given time interval [0,T]. If I could find that, I might be able to find the answer to my CTMC question.