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## Ask Dr. Arthur Rubin Your Own Question

Dr. Arthur Rubin, Professional w/Adv. Degree
Category: Homework
Satisfied Customers: 1492
Experience:  Ph.D. in Mathematics, 1978, from the California Institute of Technology, over 20 published papers
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### Customer Question

For Matrix p shown below, determine the long-run behavior of successive state matrices from the inititial state matrix [0.1 0.2 0.7 0]

A B C D
matrix p A 1 0 0 0
B 0 1 0 0
C 0.1 0.2 0.6 0.1
D 0.15 0.15 0.35 0.35
Submitted: 4 years ago.
Category: Homework
Expert:  Dr. Arthur Rubin replied 4 years ago.

Dr. Arthur Rubin :

If you are asking what P^n looks like for large n, the answer is:

Dr. Arthur Rubin :

A + B (b^n) + C (c^n), where:

Dr. Arthur Rubin :

A = {1 0 0 0 | 0 1 0 0 | 16/45 29/45 0 0 | 19/45 26/45 0 0 }

Dr. Arthur Rubin :

and b and c are less than 1

Dr. Arthur Rubin :

I can produce exact values in about 6 hours, if you are interested, but this gives the long-run approximation.

Dr. Arthur Rubin :

It took less time than I thought

Dr. Arthur Rubin :

b = 0.7 (7/10)

Dr. Arthur Rubin :

B = {0 0 0 0 | 0 0 0 0 | -10/27 -17/27 7/9 2/9 | -10/27 -17/27 7/9 2/9}

Dr. Arthur Rubin :

c = 0.25 (1/4)

Dr. Arthur Rubin :

C = { 0 0 0 0 | 0 0 0 0 | 2/135 -2/135 2/9 -2/9 | -7/135 7/135 -7/9 7/9}

Dr. Arthur Rubin :

writing x = [0.1 0.2 0.7 0], we get P^n x is

Dr. Arthur Rubin :

[0.1 + (16/45)0.7 0.2+(29/45)0.7 0 0] + b^n 0.7 [-10/27 -17/27 7/9 2/9] + c^n 0.7 [2/135 -2/135 2/9 -2/9]

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