An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 20% 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?
Poor= % (round to the nearest tenth if necessary.)
Satisfactory= % (round to the nearest tenth if necessary.)
Preferred= % (round to the nearest tenth if necessary.)
Work:
Consider the transition from one year to the next: If the poor customers (p) become satisfactory (s) in 20% of the cases and never become preferred (e), it means they stay poor in 80% of the cases
Similarly, if s moves to p 5% of the times and to e 5% of the times, it means they stay s 90% of the times
Finally, if e never moves to p and moves to s 5% of the times, it means they stay s 95% 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 e
each category is therefore
A = [ 0.80 0.05 0]
[ 0.20 0.90 0.05]
[ 0 0.05 0.95]
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 =
[ 0.05 0.05 0.05]
[0.20 0.20 0.20]
[0.20 0.20 0.20]
As p+s+e =1 for any year (as the percentages in each category have to add up to 1) if follows that
v(7) = ( 0.05 , 0.20, 0.20) independently of what v(1) is and
v(n) = v(7) for any n>=7.
So in the long run there will be 10% of poor customers, 40% of satisfactory customers and 40% of preferred customers
This was incorrect the answer for poor should be 11.1% can you help my solve for the other two and find errors
Jon L.
02/13/14