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.
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 , 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.
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).