The claim frequency, [math]N[/math], for a policy has a binomial distribution with the following properties:

Determine [math]\operatorname{P}[N=4][/math].

• 0.0301
• 0.0329
• 0.0412
• 0.0465
• 0.0493

An insurer sells an annual policy with a claim frequency with the following distribution:

The parameter [math]q[/math] is unknown but your prior beliefs indicate that it has the following density function on [0,1]:

Determine the expected annual claim frequency.

1. 0.6
2. 2.3
3. 2.4
4. 3.25
5. 4

An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. The number of claims filed has a Poisson distribution.

Calculate the variance of the number of claims filed.

• [math]\frac{1}{\sqrt{3}}[/math]
• 1
• [math]\sqrt{2}[/math]
• 2
• 4

A company prices its hurricane insurance using the following assumptions:

1. In any calendar year, there can be at most one hurricane.
2. In any calendar year, the probability of a hurricane is 0.05.
3. The numbers of hurricanes in different calendar years are mutually independent.

Using the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 20-year period.

• 0.06
• 0.19
• 0.38
• 0.62
• 0.92

An insurance policy on an electrical device pays a benefit of 4000 if the device fails during the first year. The amount of the benefit decreases by 1000 each successive year until it reaches 0. If the device has not failed by the beginning of any given year, the probability of failure during that year is 0.4.

Calculate the expected benefit under this policy.

• 2234
• 2400
• 2500
• 2667
• 2694

A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 100 (ticket cost + 50 penalty) to the tourist. Calculate the expected revenue of the tour operator.

• 955
• 962
• 967
• 976
• 985

An electronic system contains three cooling components that operate independently. The probability of each component’s failure is 0.05. The system will overheat if and only if at least two components fail.

Calculate the probability that the system will overheat.

• 0.007
• 0.045
• 0.098
• 0.135
• 0.143

The number of days an employee is sick each month is modeled by a Poisson distribution with mean 1. The numbers of sick days in different months are mutually independent.

Calculate the probability that an employee is sick more than two days in a three-month period.

• 0.199
• 0.224
• 0.423
• 0.577
• 0.801

Company XYZ provides a warranty on a product that it produces. Each year, the number of warranty claims follows a Poisson distribution with mean [math]c[/math]. The probability that no warranty claims are received in any given year is 0.60. Company XYZ purchases an insurance policy that will reduce its overall warranty claim payment costs. The insurance policy will pay nothing for the first warranty claim received and 5000 for each claim thereafter until the end of the year.

Calculate the expected amount of annual insurance policy payments to Company XYZ.

• 554
• 872
• 1022
• 1354
• 1612

A life insurance company has found there is a 3% probability that a randomly selected application contains an error. Assume applications are mutually independent in this respect. An auditor randomly selects 100 applications.

Calculate the probability that 95% or less of the selected applications are error-free.

• 0.08
• 0.10
• 0.13
• 0.15
• 0.18