Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose $p=\begin{bmatrix} 0& 1\over 3 &0 &2\over 3 \\ 0.3& 0& 0.7 &0 \\ 0& 2\over 3&0 &1\over3 \\ 0.8& 0& 0.2& 0 \end{bmatrix}$is the transition probability matrix of a Markov chain with state space {1, 2, 3, 4}. How to get the limiting probabilities of the Markov chain. I think if more different alternative answer and apporach would be better. In fact i know a little bit about limiting probability but not sure how to apply in this question. Clear step to illustrate how it works or explaination would be appreciated and i wanna learn how others interpret the concept of probability

share|improve this question
    
Note that each state has period two, since if you start in positions $1$ or $3$, then after one step you will be at $2$ or $4$ and then after another back at $1$ or $3$. –  Henry Nov 20 '12 at 7:42

1 Answer 1

To find an invariant distribution for a Markov chain, which will give you information about the long term probability, you can use two methods. Solving the "left hand equations" for an irreducible, recurrent Markov chain: $$\pi_i = \displaystyle \sum_{j\in I} P_{ji} \pi_j$$

(Where $I$ is the state space)will give you an invariant measure, and the restriction

$$\displaystyle \sum_{j\in I} \pi_j = 1$$

will give you an invariant distribution. These equations give the long term proportion of time spent in each state $i$ as $\pi_i$. This is invariant because it is saying "the probability of being in a state $i$ is the same as the sum of the probabilities being in any other state $j$ (which is $\pi_j$) and then moving into state $i$ (which is $P_{ji}$)

When you're more comfortable with the idea of invariant distributions, you can often save time (as you can in this case) by looking at the so-called "detailed balance equations", but they're more complicated, and I've run out of time! It's best to start with the basics though, and look up more info on detailed balance a bit later if you ask me.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.