For a certain health insurance policy, losses are uniformly distributed on the interval [0, [math]b[/math]]. The policy has a deductible of 180 and the expected value of the unreimbursed portion of a loss is 144.

Calculate [math]b[/math].

• 236
• 288
• 388
• 450
• 468

The working lifetime, in years, of a particular model of bread maker is normally distributed with mean 10 and variance 4. Calculate the 12th percentile of the working lifetime, in years.

• 5.30
• 7.65
• 8.41
• 12.35
• 14.70

An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2.

Calculate the expected claim payment made for an auto loss.

• [math]0.5e^{−2} − 0.5e^{−12}[/math]
• [math]2e^{-1/2}-7e^{-3}[/math]
• [math]2e^{-1/2}-2e^{-3}[/math]
• [math]2e^{-1/2}[/math]
• [math]3e^{-1/2}-2e^{-3}[/math]

The loss [math]L[/math] due to a boat accident is exponentially distributed. Boat insurance policy A covers up to 1 unit for each loss. Boat insurance policy B covers up to 2 units for each loss. The probability that a loss is fully covered under policy B is 1.9 times the probability that it is fully covered under policy A.

Calculate the variance of [math]L[/math].

• 0.1
• 0.4
• 2.4
• 9.5
• 90.1

An insurance policy is written to cover a loss, [math]X[/math], where [math]X[/math] has a uniform distribution on [0, 1000]. The policy has a deductible, [math]d[/math], and the expected payment under the policy is 25% of what it would be with no deductible.

Calculate [math]d[/math].

• 250
• 375
• 500
• 625
• 750

An automobile insurance company issues a one-year policy with a deductible of 500. The probability is 0.8 that the insured automobile has no accident and 0.0 that the automobile has more than one accident. If there is an accident, the loss before application of the deductible is exponentially distributed with mean 3000.

Calculate the 95th percentile of the insurance company payout on this policy.

• 3466
• 3659
• 4159
• 8487
• 8987

An insurer offers a travelers insurance policy. Losses under the policy are uniformly distributed on the interval [0, 5]. The insurer reimburses a policyholder for a loss up to a maximum of 4.

Determine the cumulative distribution function, [math]F[/math], of the benefit that the insurer pays a policyholder who experiences exactly one loss under the policy.

• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.20x, \, 0 \leq x \lt 4 \\ 1, \, x \geq 4 \end{cases}[[/math]]
• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.20x, \, 0 \leq x \lt 5 \\ 1, \, x \geq 5 \end{cases}[[/math]]
• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.25x, \, 0 \leq x \lt 4 \\ 1, \, x \geq 4 \end{cases}[[/math]]
• [[math]]F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.25x, \, 0 \leq x \lt 5 \\ 1, \, x \geq 5 \end{cases}[[/math]]
• [[math]] F(x) = \begin{cases} 0, \, x \lt 0 \\ 0.25x, \, 1 \leq x \lt 5 \\ 1, \, x \geq 5 \end{cases} [[/math]]

A car and a bus arrive at a railroad crossing at times independently and uniformly distributed between 7:15 and 7:30. A train arrives at the crossing at 7:20 and halts traffic at the crossing for five minutes. Calculate the probability that the waiting time of the car or the bus at the crossing exceeds three minutes.

• 0.25
• 0.27
• 0.36
• 0.40
• 0.56

(Ross^{[Notes 1]}) An expert witness in a paternity suit testifies that the length (in days) of a pregnancy, from conception to delivery, is approximately normally distributed, with parameters [math]\mu = 270[/math], [math]\sigma = 10[/math]. The defendant in the suit is able to prove that he was out of the country during the period from 290 to 240 days before the birth of the child. What is the probability that the defendant was in the country when the child was conceived?

• 0.0195
• 0.02
• 0.0221
• 0.0231
• 0.0241

Notes

References

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

Let [math]X[/math] and [math]Y[/math] be independent random variables with uniform density functions on [math][0,1][/math]. Find [math]E((X + Y)^2)[/math].

• 2/3
• 3/4
• 1
• 7/6
• 3/2

References

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