Markov Question

Question

An auto insurance company classifies its customers in these categories poor, satisfactory, and preferred. Each year 15% of those in the poor category are moved to satisfactory and 5% of those in the satisfactory category are moved to preferred. Also, 5% in the preferred category are moved to the satisfactory category, and 5% of those in the satisfactory category are moved to the poor category. Customers are never moved from poor to preferred, or conversely, in a single year. Assuming these percentages remain valid over a long period of time, how many customers can the company expect to have in each category in the long run?

Answer 1

We find the transition matrix, then the eigenvector associated with the eigenvalue of 1. The eigenvector is "normalized" to sum to 1. Let poor be x, satisfactory be y, and preferred be z

P =

.85 .15 0

.05 .9 .05

0 .05 .95

Let z = 1

From the third column, .05y + .95 = 1; .05y = .05; y = 1

From the first column, .85x + .05 = x

.05 = .15x

x = 1/3

Then, we normalize. x + y + z = 7/3; therefore.

x = 1/3* 3/7 = 1/7

y = 1 * 3/7 = 3/7

z = 1 * 3/7 = 3/7

Poor = 1/7; Satisfactory = 3/7; Preferred = 3/7