I have been unable to solve the following exercise from my Probability and Statistics book.
I succeeded in expressing observations (ii) and (iii) as mathematical statements:Under an insurance policy, a maximum of five claims may be filed per year by a policyholder. Let be the probability that a policyholder files n claims during a given year, where = 1,2,3,4,5. An actuary makes the following observations:
(i) for = 1,2,3,4.
(ii) The difference between and is the same for = 1,2,3,4.
(iii) Exactly 40% of the policyholders file fewer than three claims during a given year.
Calculate the probability that a random policyholder will file more than three claims during a given year.
(A) 0.14 (B) 0.16 (C) 0.27 (D) 0.29 (E) 0.33
(ii) for = 1,2,3,4.
(iii)
I also know the probabilities sum to 1, so
To solve the exercise, I need to find , but I have been unable to do so. I tried rewriting the sum of probabilities like so:
I have tried other approaches, but it seems I am aimlessly solving equations that are just restatements of each other. I believe if I could find and , then I could find and . If anyone could point me in the right direction, I would appreciate it.
By the way, the solutions manual says the answer is (C) 0.27.
Thank you for your reply. In [2], it appears you made the substitution , but only applies for . Does this change anything?
Since it doesn't apply to , a possible distribution satisfying the conditions is as follows , , , , , in which case the answer is 0.43.
the relationship applies to n=1,2,3,4,5.
You already noted that (n=1,2,3,4)
use n=4, so:
and then use the same reasoning as before.
Remark
If you believe that the above isn't valid for then there are infinately many solutions to the problem(this isn't hard to show).