If [math]X[/math] has a normal distribution with mean 2 and standard deviation 1.5, determine [math]\operatorname{P}[X^2 - X -2 \gt 0 ] [/math].

• 0
• 0.477
• 0.5
• 0.523
• 1

An actuary is using a normal approximation to model loss distributions. To calibrate the parameters of the normal distribution, the actuary uses historical loss data. You are given the following:

• The sample mean equals 500
• 10% of the historical losses are larger than 800

Approximate the 95^th percentile of the loss distribution.

1. 678.15
2. 750.33
3. 856.11
4. 884.96
5. 900.25

An insurance policy reimburses dental expense, [math]X[/math], up to a maximum benefit of 250. The probability density function for [math]X[/math] is:

where [math]c[/math] is a constant. Calculate the median benefit for this policy.

• 161
• 165
• 173
• 182
• 250

The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year.

Calculate the portion of high-risk drivers are expected to be involved in an accident during the first 80 days of a calendar year.

• 0.15
• 0.34
• 0.43
• 0.57
• 0.66

The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 250. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (0, 1500).

Calculate the standard deviation of the insurance payment in the event that the automobile is damaged.

• 361
• 403
• 433
• 464
• 521

An insurance company sells an auto insurance policy that covers losses incurred by a policyholder, subject to a deductible of 100. Losses incurred follow an exponential distribution with mean 300.

Calculate the 95th percentile of losses that exceed the deductible.

• 600
• 700
• 800
• 900
• 1000

In 1982 Abby’s mother scored at the 93rd percentile in the math SAT exam. In 1982 the mean score was 503 and the variance of the scores was 9604. In 2008 Abby took the math SAT and got the same numerical score as her mother had received 26 years before. In 2008 the mean score was 521 and the variance of the scores was 10,201. Math SAT scores are normally distributed and stated in multiples of ten.

Calculate the percentile for Abby’s score.

• 89th
• 90th
• 91st
• 92nd
• 93rd

A car is new at the beginning of a calendar year. The time, in years, before the car experiences its first failure is exponentially distributed with mean 2.

Calculate the probability that the car experiences its first failure in the last quarter of some calendar year.

• 0.081
• 0.088
• 0.102
• 0.205
• 0.250

An insurance company issues policies covering damage to automobiles. The amount of damage is modeled by a uniform distribution on [0, b]

The policy payout is subject to a deductible of b/10.

A policyholder experiences automobile damage. Calculate the ratio of the standard deviation of the policy payout to the standard deviation of the amount of the damage.

• 0.8100
• 0.9000
• 0.9477
• 0.9487
• 0.9735

Losses due to burglary are exponentially distributed with mean 100. The probability that a loss is between 40 and 50 equals the probability that a loss is between 60 and [math]r[/math], with [math]r \gt 60[/math]. Calculate [math]r[/math].

• 68.26
• 70.00
• 70.51
• 72.36
• 75.00