# Find the limiting Distribution of 2-state transition matrix.

Consider the two-state Marcov chain with transition matrix $$P= \ \begin{pmatrix} 1-p & p \\ 1 & 0 \end{pmatrix}$$

Find the limiting Distribution.

Let $\pi=(\pi_1,\pi_2)$ be the distribution vector.

Then , we have

$\pi P=\pi$ which gives us

$\pi_1 (1-p)+ \pi_2=\pi_1, \\ \pi_1p+0=\pi_2,$

Solving we get

$p=0$

Thus,

$\pi_2=0 , \ \pi_1=constant=c, say$

But since $\pi_1+\pi_2=1$ , we see $\pi_1=1, \ \pi_2=0$

Thus the limiting distribution is $\ \pi=(1,0)$

Am I right?

$$\pi_2 = p\pi_1$$
$$\pi_1 + \pi_2=1$$
$$(p+1)\pi_1=1$$
$$\pi_1=\frac1{p+1}$$
$$\pi_2=\frac{p}{1+p}$$
Note that $p$ is not a variable to be solved, it is fixed and it is given to you.