# Math Help - Somewhat simple actuary question about filing claims.

1. ## Somewhat simple actuary question about filing claims.

Under an insurance policy, a maximum of …five claims may be filed per year by a policyholder. Let $p_n$ be the probability that a policyholder files n claims during a given year, where n = 0,1,2,3,4,5. An actuary makes
the following observations:

(a) $p_n$ >= $p_{n+1}$ for n = 0, 1, 2, 3, 4.
(b) The difference between $p_n$ and $p_{n+1}$ is the same for n = 0, 1, 2, 3, 4.
(c) Exactly 40% of policyholders file fewer than two claims during a given year.

How would you calculate the probability that a random policyholder will …file more than three claims during a given year?

2. The constraints imply that the shape of the distribution must be trapezoidal.
You need to find the 2 key values on the y-axis to get the pdf, say k1 and k2.

Now you have 2 equations and 2 unknowns: 1 equation is provided by the fact that the total probability=1.
The 2nd is provided by constraint (c).

Solve simultaneously for k1 and k2.
Now the required probability is f(4) + f(5) where f is the pdf.

3. Thanks