Find the long-run probability of the system being in state 1?

Question

P= [ 1/4 3/4 ]
[ 2/3 1/3 ] is the transition matrix for a regular Markov Chain. (line the two brackets up one under the other)

Answer 1

Hi, so we're looking for the long-run solution here.
We have two states, since it is a 2x2 transition matrix.
Let the probabilites for being in each state form a vector, call it x. So, xP = x or x(P-I) = 0.
Thus,
x times
[-3/4 3/4]
[2/3 -2/3] = 0.
So, we have one equation that x1(-3/4) +x2(2/3) = 0.
But don't forget that there are only two states and x1 and x2 are probabilities, so x1 + x2 = 1. Solve this system of equations for x1 and you get: x1 = 8/17 and x2 = 9/17.

Therefore, the long run probability of the system being in state 1 is 8/17.

Answer 2

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