# Thread: Transition Matrix - Stationary Distribution

1. ## Transition Matrix - Stationary Distribution

Consider the one-step transition matrix

$\displaystyle P = \begin{pmatrix} q & p & 0 \\0 & q & p \\ p & 0 & q \\ \end{pmatrix}$

where $\displaystyle q = 1 - p$ and $\displaystyle 0 < p < 1$.

Find all stationary distributions of a chain with state space $\displaystyle E = \{0,1,2\}$ and one-step transition matrix P.

Are any stationary distributions also limiting distributions?

Thanks

2. My mid-term is in two days and I really need to understand how to calculate this sort of question.

I have.....
The equation $\displaystyle \pi = \pi P$ becomes

$\displaystyle \pi_0 = q \pi_0 + p \pi_1$

$\displaystyle \pi_1 = q \pi_1 + p \pi_2$

$\displaystyle \pi_2 = p \pi_0 + q \pi_2$

I really dont know how to get the solutions from here.

3. i think,

$\displaystyle \pi_0 + \pi_1 + \pi_2 = 1$

so you can just make one of the $\displaystyle \pi$ the subject and then sub it into the 3 equations that you found yeah and continue until you found all the values of $\displaystyle \pi$

4. I think my talent for mathematics has just about run its course

I get

$\displaystyle \pi_2 = \frac{p-p\pi_1}{1-p+q}$, $\displaystyle \pi_1 = \frac{\pi_2 - p - q\pi_2}{1-q-p}$ and $\displaystyle \pi_0 = \frac{\pi_1 - q\pi_1 - \pi_2 +p+q\pi_2}{1-q}$

Im confused with the difference between the limiting distribution and the stationary distribution???? Can anyone please explain this?