Let [math]X[/math] denote the loss amount sustained by an insurance company’s policyholder in an auto collision. Let [math]Z[/math] denote the portion of [math]X[/math] that the insurance company will have to pay. An actuary determines that [math]X[/math] and [math]Z[/math] are independent with respective density and probability functions

and

Calculate the variance of the insurance company’s claim payment [math]ZX[/math].

• 13.0
• 15.8
• 28.8
• 35.2
• 44.6

Claim amounts at an insurance company are independent of one another. In year one, claim amounts are modeled by a normal random variable X with mean 100 and standard deviation 25. In year two, claim amounts are modeled by the random variable [math]Y = 1.04X + 5[/math].

Calculate the probability that a random sample of 25 claim amounts in year two average between 100 and 110.

• 0.48
• 0.53
• 0.54
• 0.67
• 0.68

A delivery service owns two cars that consume 15 and 30 miles per gallon. Fuel costs 3 per gallon. On any given business day, each car travels a number of miles that is independent of the other and is normally distributed with mean 25 miles and standard deviation 3 miles.

Calculate the probability that on any given business day, the total fuel cost to the delivery service will be less than 7.

• 0.13
• 0.23
• 0.29
• 0.38
• 0.47

A number [math]U[/math] is chosen at random in the interval [math][0,1][/math]. Find the probability that [math]T = U/(1 - U) \lt 1/4[/math].

• 1/6
• 1/5
• 1/4
• 1/2
• 2/3

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

An insurance policy offers coverage for three years. You are given the following about the annual loss:

• Annual losses are independent
• The annual loss has an exponential distribution with mean 500

Determine the mode of the loss distribution for the three year coverage.

• 500
• 750
• 1,000
• 2,000
• 2,500

In each of the months June, July, and August, the number of accidents occurring in that month is modeled by a Poisson random variable with mean 1. In each of the other 9 months of the year, the number of accidents occurring is modeled by a Poisson random variable with mean 0.5. Assume that these 12 random variables are mutually independent.

Calculate the probability that exactly two accidents occur in July through November.

• 0.084
• 0.185
• 0.251
• 0.257
• 0.271

In a given region, the number of tornadoes in a one-week period is modeled by a Poisson distribution with mean 2. The numbers of tornadoes in different weeks are mutually independent.

Calculate the probability that fewer than four tornadoes occur in a three-week period.

• 0.13
• 0.15
• 0.29
• 0.43
• 0.86

A certain brand of refrigerator has a useful life that is normally distributed with mean 10 years and standard deviation 3 years. The useful lives of these refrigerators are independent.

Calculate the probability that the total useful life of two randomly selected refrigerators will exceed 1.9 times the useful life of a third randomly selected refrigerator.

• 0.407
• 0.444
• 0.556
• 0.593
• 0.604

Claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000.

Calculate the probability that the average of 25 randomly selected claims exceeds 20,000.

• 0.01
• 0.15
• 0.27
• 0.33
• 0.45

The random variables [math]X,Y[/math] have the joint density function

for a constant [math]c[/math]. Determine the joint density function for the random variables [math]W = Y^{-1}, Z = X^{-1}.[/math]

• [[math]] f_{W,Z}(w,z) = \begin{cases} c wz \, e^{-1/wz}, 0 \lt w \lt z \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] f_{W,Z}(w,z) = \begin{cases} c wz \, e^{-w/z}, 0 \lt w \lt z \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]]f_{W,Z}(w,z) = \begin{cases} c wz \, e^{-wz}, 1 \lt z \lt w \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f_{W,Z}(w,z) = \begin{cases} c w^2z \, e^{-1/(wz)}, 0\lt z \lt w \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f_{W,Z}(w,z) = \begin{cases} c wz^2 \, e^{-1/(wz)}, 0\lt z \lt w \lt 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]