An auto insurance company classifies its customers in three categories?

An auto insurance company classifies its customers in three categories?

An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 35% of those in the poor category are moved to satisfactory and 30% of those in the satisfactory category are moved to preferred. Also 30% in the preferred category are moved to the satisfactory category and 30% 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?

Ana

Consider the transition from one year to the next: If the poor customers (p) become satisfactory (s) in 35% of the cases and never become preferred (e), it means they stay poor in 65% of the casesSimilarly, if s moves to p 30% of the times and to e 30% of the times, it means they stay s 40% of the timesFinally, if e never moves to p and moves to s 30% of the times, it means they stay s 70% of the times.The transition matrix from v(1)=(p,s,e) in year 1 to v(2)=(p,s,e) in Year 2, if p, s and e represent the percentages in each category is therefore A =[ 0.65 0.3 0][ 0.35 0.4 0.3][ 0 0.3 0.7]If v(2) = A v(1), then v(3) = A v(2) = A^2 v(1) and v(n) = A^n v(1). Calculating A^n for a few values of n we will see that A^7 =[ 0.3 0.3 0.3][0.35 0.35 0.35][0.35 0.35 0.35]As p+s+e =1 for any year (as the percentages in each category have to add up to 1) if follows thatv(7) = ( 0.3 , 0.35, 0.35) independently of what v(1) is andv(n) = v(7) for any n>=7. So in the long run there will be 30% of poor customers, 35% of satisfactory customers and 35% of preferred customers

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