For a collective risk model the number of losses, N, has a Poisson distribution with [math]\lambda = 20[/math]. The common distribution of the individual losses has the following characteristics:

1. [math]\operatorname{E}[ X ] = 70 [/math]
2. [math]\operatorname{E}[ X \wedge 30] = 25 [/math]
3. [math]\operatorname{Pr}( X \gt 30) = 0.75 [/math]
4. [math]\operatorname{E}[ X^2 | X \gt 30] = 9000 [/math]

An insurance covers aggregate losses subject to an ordinary deductible of 30 per loss.

Calculate the variance of the aggregate payments of the insurance.

• 54,000
• 67,500
• 81,000
• 94,500
• 108,000

For a collective risk model:

1. The number of losses has a Poisson distribution with [math]\lambda = 2 [/math].
2. The common distribution of the individual losses is:

An insurance covers aggregate losses subject to a deductible of 3.

Calculate the expected aggregate payments of the insurance.

• 0.74
• 0.79
• 0.84
• 0.89
• 0.94

For an insurance:

1. The number of losses per year has a Poisson distribution with [math]\lambda = 10 [/math] .
2. Loss amounts are uniformly distributed on (0, 10).
3. Loss amounts and the number of losses are mutually independent.
4. There is an ordinary deductible of 4 per loss.

Calculate the variance of aggregate payments in a year.

• 36
• 48
• 72
• 96
• 120

For the workers’ compensation claims of a construction company you are given:

1. The annual number of claims follows the Poisson distribution with mean 20.
2. Claim sizes X follow the lognormal distribution with [math]\mu = 4.2 [/math] and [math]\sigma = 1.1[/math] .
3. The company retains the first 500 of each claim.
4. Annual aggregate retained claims approximately follow the normal distribution.
5. [math]\operatorname{E}[(X \wedge 500)^2] = 26189[/math]

Determine the 90th percentile of the aggregate distribution of retained claims.

• Less than 2900
• At least 2900, but less than 3100
• At least 3100, but less than 3300
• At least 3300, but less than 3500
• At least 3500

1. Loss amounts and claim numbers are independent within and between products.



2. 		Product X	Product Y
   Number of Claims	Mean	10	2
   	Standard Deviation	3	1
   Loss Amount	Mean	20	50
   	Standard Deviation	5	10




3. Aggregate losses, for both products combined, approximately follow the normal distribution.

Determine the probability that aggregate losses for both products combined exceed 400.

• Less than 0.10
• At least 0.10, but less than 0.15
• At least 0.15, but less than 0.20
• At least 0.20, but less than 0.25
• At least 0.25

Computer maintenance costs for a department are modeled as follows:

1. The distribution of the number of maintenance calls each machine will need in a year is Poisson with mean 3.
2. The cost for a maintenance call has mean 80 and standard deviation 200.
3. The number of maintenance calls and the costs of the maintenance calls are all mutually independent.

The department must buy a maintenance contract to cover repairs if there is at least a 10% probability that aggregate maintenance costs in a given year will exceed 120% of the expected costs.

Calculate the minimum number of computers needed to avoid purchasing a maintenance contract using the normal approximation for the distribution of the aggregate maintenance costs.

• 80
• 90
• 100
• 110
• 120

At the beginning of each round of a game of chance the player pays 12.5. The player then rolls one die with outcome N. The player then rolls N dice and wins an amount equal to the total of the numbers showing on the N dice. All dice have 6 sides and are fair.

Calculate the probability that a player starting with 15,000 will have at least 15,000 after 1000 rounds using the normal approximation.

• 0.01
• 0.04
• 0.06
• 0.09
• 0.12

For a certain company, losses follow a Poisson frequency distribution with mean 2 per year, and the amount of a loss is 1, 2, or 3, each with probability 1/3. Loss amounts are independent of the number of losses, and of each other.

An insurance policy covers all losses in a year, subject to an annual aggregate deductible of 2.

Calculate the expected claim payments for this insurance policy.

• 2.00
• 2.36
• 2.45
• 2.81
• 2.96

A dam is proposed for a river that is currently used for salmon breeding. You have modeled:

• For each hour the dam is opened the number of salmon that will pass through and reach the breeding grounds has a distribution with mean 100 and variance 900.
• The number of eggs released by each salmon has a distribution with mean 5 and variance 5.
• The number of salmon going through the dam each hour it is open and the numbers of eggs released by the salmon are independent.

Calculate the least number of whole hours the dam should be left open so the probability that 10,000 eggs will be released is greater than 95% using the normal approximation for the aggregate number of eggs released.

• 25
• 23
• 26
• 29
• 32

The number of claims, N, made on an insurance portfolio follows the following distribution:

If a claim occurs, the benefit is 0 or 10 with probability 0.8 and 0.2, respectively.

The number of claims and the benefit for each claim are independent.

Calculate the probability that aggregate benefits will exceed expected benefits by more than 2 standard deviations.

• 0.02
• 0.05
• 0.07
• 0.09
• 0.12