Answered: 3 -2 3 I2N 120 180 90

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 14E

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Question

let P be the transition matrix for a Markov chain with three states.Let x0 be the initial state vector for the population.Compute Find the steady state vector

Expert Solution

Step 1

a ] the state vector and is given by ,

so we get ,

Step 2

now is given by

$P^2=\begin{bmatrix} 1/2 &1/3 &1/3 \\ 0 & 1/3 & 2/3\\ 1/2 &1/3 &0 \end{bmatrix}\begin{bmatrix} 1/2 &1/3 &1/3 \\ 0 & 1/3 & 2/3\\ 1/2 &1/3 &0 \end{bmatrix}=\begin{bmatrix} 5/12 &7/18 & 7/18\\ 1/3 & 1/3 &2/9 \\ 1/4 & 5/18 &7/18 \end{bmatrix}$

$X_2=P^2X_0=\begin{bmatrix} 5/12 &7/18 & 7/18\\ 1/3 & 1/3 &2/9 \\ 1/4 & 5/18 &7/18 \end{bmatrix}\begin{bmatrix} 120\\ 180\\ 90 \end{bmatrix}=\begin{bmatrix} 120(5/12)+180(7/18)+90(7/18)\\ 120(1/3)+180(1/3)+90(2/9)\\ 120(1/4)+180(5/18)+90(7/18) \end{bmatrix}$

we get as ,

${\color{DarkBlue} \mathbf{X_2=\begin{bmatrix} 155\\ 120\\ 115 \end{bmatrix}}}$

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