Under an insurance policy, a maximum of 5 claims may be filed per year by a poli

Question

Under an insurance policy, a maximum of 5 claims may be filed per year by a policyholder. Let X be the random variable representing the number of claims that a policyholder will file. X=0,l,2,3,4,5. Let p(x) be the probability that a policyholder files x claims during a given year, where x=0,l,2,3,4,5. An actuary makes the following observations. The differences between consecutive probabilities are the same. In other words..... Exactly 40% of the policyholders file less that 2 claims during a year. Calculate the probability that a random policyholder will file more than 3 claims during a given year. a..14 b..16 c..27 d..29 e.33

Explanation / Answer

We know that p0 + p1 = 2 5 = 0.4

p0 + p1 + p2 + p3 + p4 + p5 = 1.

k = p0 p1 = p1 p2 = p2 p3 = p3 p4 = p4 p5 .

Then p1 = p0 k, p2 = p1 k = p0 k k = p0 2k, etc.,

so that pn = p0 nk for 1 n 5.

Thus p0 + ( p0 k) + ( p0 2k) +…+ ( p0 5k) = 6 p0 15k = 1.

Also 40% = 0.4 = p0 + p1 = p0 + ( p0 k) = 2 p0 k.

This results in the following equations:

6 p0 15k = 1,

2 p0 k = 0.4,

Solving, the above: k = 1/60 and p0 = 25 /120 .

Therefore: p4 + p5 = ( p0 4k) + ( p0 5k)

= 17 /120 + 15 /120

= 32 /120 = 0.2667 ~ 0.27

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