Question

Please apply Linear Algebra. Non-calculator. Thanks

7. Devon will only wear red, green, or blue shirts. Suppose that a Markov chain is established for Devon's daily wardrobe choices and suppose that the state vectors are apportioned as follows. probability that Devon wears a blue shirt on "day k" probability that Devon wears a green shirt on "day k" V, k probability that Devon wears a red shirt on "day k" The actual questions appear on the next page. 75T 75 2 Suppose further that the transition matrix for the Markov chain is 4- 25 25 25 What is the contextual significance of the entry a,0.75? (1 point) a. Determine the stochastic steady state vector for the Markov chain. Show all relevant work and b make sure that your conclusion is clear. (8 points) What is the contextual significance of the bottom entry of the stochastic steady state vectorw C. found in part (b) of the question? (1 point)

Answer 1

Answer:

a) Contextual significance of value is as follows,

a12 0.75

It is the probability of devon wearing red shirt next day provided that currently devon is wearing blue shirt which is 0.75.

b) The process to compute stochastic steady state vector is as follows:

1) Take any eigen vector of A with eigen value 1 by solving characteristic equation.

2) Divide the vector by sum of vector to make sum of all entries as 1

3) Vector v is the stochastic steady state vector

Let's apply the steps to compute the stochastic steady state vector,

Characteristic polynomial is

  

А - A %3D0 ГО.5 — А 0.75 0.75 —А 0.25|| — 0 0.25 - 0.25 0.25

  

(0.5 (20.0625) 0.75(-0.25A -0.0625) +0.75(0.0625+0.25A) 0

(0.5 A(A0.25) A0.25)0.1875( 0.25)0.1875 (0.25 )=0

  

A0.25)(0.5- A)(a - 0.25)0.375 0 (A0.25)- 0.75 0.125+0.375] (A0.25)- +0.750.25]0 (入+0.25)[ -0.75入- 0.25] = 0 0

  

A0.25)A 1)(A 0.25) 0 = 1,-0.25

Let's find out the eigen vector for eigen value 1,

  

AX(1)I

  

0.5 0.75 0.75 = I 0.25 0.25 0 2 0.25 0.25 0 3

Solving above equation we get,

  

12.5, x2 = 1.5, x3 = 1.5

so eigen vector is ,

  

2.5 1.5 1.5

and after dividing by sum of elements we get stochastic vector,

  

2.5 1 1.5 5.515

c) The bottom entry is 1.5/5.5 = 0.272 = 27.2 % which says that probability of devon wearing red shirt is 27.2%