What is the formula to find average duration of state s in a Markov chain given a transition matrix?

I tried to recall the concept but could not find any references.


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I understand that we are talking about a discrete time process. If the probability of staying in state $s$ from one period to another is $$P(X_n=s|X_{n-1}=s)=q,\quad \forall n\ge1,$$ then the number $N$ of consecutive repetitions of the state $s$ is a random variable with geometric distribution, that is $$N\sim \mathcal G(1-q),$$ since every transition can be seen as a dicotomic experiment where the "fail" consists in staying at state $s$ (with probability $1-q$) and "success" is changing to any other state (with probability $q$).

In fact, there are several alternative definitions of a geometric r.v., but here I consider it as the number of "fail" results before the first "success". In this case, we have $E(N)=\frac q{1-q}$.

Since the actual number of successive $s$ states would be $N+1$, if we consider the first time we reach that state, the expected number of $s$ states once the chain reaches it would be $$E(N+1)=1+\frac q{1-q}=\frac1{1-q},$$ counting the current time, or $$E(N)=\frac q{1-q},$$ if we count from next time on.

  • $\begingroup$ I have an example of a transition matrix P = (7/8, 1/8; 1/8, 7/8) and it says that the expected time in each of the two states is 8 periods... I cannot figure out the calculations... $\endgroup$ – zmicer Jan 25 at 6:48
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    $\begingroup$ Take $q=7/8$ and you'll get the result. $\endgroup$ – Alejandro Nasif Salum Jan 25 at 6:52
  • $\begingroup$ @ math.stackexchange.com/users/481187/alejandro-nasif-salum :) Thanks! $\endgroup$ – zmicer Jan 25 at 6:55

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