You are given:

Claim sizes are independent.

Calculate the variance of the aggregate loss.

• 4,050
• 8,100
• 10,500
• 12,510
• 15,612

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

Aggregate losses for a portfolio of policies are modeled as follows:

1. The number of losses before any coverage modifications follows a Poisson distribution with mean [math]\lambda[/math] .
2. The severity of each loss before any coverage modifications is uniformly distributed between 0 and b.

The insurer would like to model the effect of imposing an ordinary deductible, d (0 < d < b) , on each loss and reimbursing only a percentage, c (0 < c < 1) , of each loss in excess of the deductible.

It is assumed that the coverage modifications will not affect the loss distribution.

The insurer models its claims with modified frequency and severity distributions. The modified claim amount is uniformly distributed on the interval [0, c(b − d )] .

Determine the mean of the modified frequency distribution.

• [math]\lambda [/math]
• [math]\lambda c[/math]
• [math] \lambda \frac{d}{b}[/math]
• [math]\lambda \frac{b-d}{b} [/math]
• [math]\lambda c \frac{b-d}{b}[/math]

A towing company provides all towing services to members of the City Automobile Club. You are given:

1. The automobile owner must pay 10% of the cost and the remainder is paid by the City Automobile Club.
2. The number of towings has a Poisson distribution with mean of 1000 per year.
3. The number of towings and the costs of individual towings are all mutually independent.

Calculate the probability that the City Automobile Club pays more than 90,000 in any given year using the normal approximation for the distribution of aggregate towing costs.

• 3%
• 10%
• 50%
• 90%
• 97%

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

A group dental policy has a negative binomial claim count distribution with mean 300 and variance 800.

Ground-up severity is given by the following table:

You expect severity to increase 50% with no change in frequency. You decide to impose a per claim deductible of 100.

Calculate the expected total claim payment after these changes.

• Less than 18,000
• At least 18,000, but less than 20,000
• At least 20,000, but less than 22,000
• At least 22,000, but less than 24,000
• At least 24,000

You own a light bulb factory. Your workforce is a bit clumsy – they keep dropping boxes of light bulbs. The boxes have varying numbers of light bulbs in them, and when dropped, the entire box is destroyed.

You are given:

• Expected number of boxes dropped per month: 50
• Variance of the number of boxes dropped per month: 100
• Expected value per box: 200
• Variance of the value per box: 400

You pay your employees a bonus if the value of light bulbs destroyed in a month is less than 8000.

Assuming independence and using the normal approximation, calculate the probability that you will pay your employees a bonus next month.

• 0.16
• 0.19
• 0.23
• 0.27
• 0.31

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

A company insures a fleet of vehicles. Aggregate losses have a compound Poisson distribution. The expected number of losses is 20. Loss amounts, regardless of vehicle type, have exponential distribution with [math] \theta = 200.[/math]

To reduce the cost of the insurance, two modifications are to be made:

1. A certain type of vehicle will not be insured. It is estimated that this will reduce loss frequency by 20%.
2. A deductible of 100 per loss will be imposed.

Calculate the expected aggregate amount paid by the insurer after the modifications.

• 1600
• 1940
• 2520
• 3200
• 3880