# Exam P Practice Problem 65 – Total Insurance Payment

Problem 65-A

The number of random losses in a calendar year for an individual has a Poisson distribution with mean 1. When a loss occurs, the individual loss amount is either 2 or 4, with probabilities 0.6 and 0.4, respectively.

When multiple losses occur for this individual, the individual loss amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 1 per loss.

In the next calendar year, let $S$ be the total payment made by the insurance company to the insured. Calculate $P(2 \le S \le 4)$.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12204$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.16974$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29380$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31689$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34277$

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Problem 65-B

The number of claims in a calendar year for an insured has a Poisson distribution with mean 1.2. When a claim occurs, the individual claim amount is either 10 or 20, with probabilities 0.8 and 0.2, respectively.

When multiple claims occur for this insured, the individual claim amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 5 per loss.

In the next calendar year, let $S$ be the total payment made by the insurance company to the insured. Calculate $P(10 \le S < 30)$.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1010$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2986$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3709$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3826$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3906$

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