Question

For the transition matrix P = [0.8 2 0.2 2, 0.3 0.7]​, solve the equation SP = S to find the stationary matrix S and the limiting matrix Upper P overbar.

Answer 1

Stationary matrix S = [ 0.6 0.4 ]

limiting matrix P =
`\left[\begin{array}{ccc}0.6&0.4\\0.6&0.4\\\end{array}\right]`

Explanation:

Transition matrix

`p = \left[\begin{array}{ccc}0.8&0.2\\0.3&0.7\\\end{array}\right]`

solving the equation SP = S ( using Markova chain with 2 states )

stationary matrix, S = [ a , 1 - a ]

given that SP = S

[ a , 1 - a ] *
`\left[\begin{array}{ccc}0.8&0.2\\0.3&0.7\\\end{array}\right]` = [ a , 1 - a ]

= a*(0.8) + ( 1 - a ) ( 0.3 ) = a

∴ a = 0.6

hence; stationary matrix S = [ 0.6 0.4 ]

limiting matrix P =
`\left[\begin{array}{ccc}0.6&0.4\\0.6&0.4\\\end{array}\right]`