At the start of a week, a coal mine has a high-capacity storage bin that is half full. During the week, 20 loads of coal are added to the storage bin. Each load of coal has a volume that is normally distributed with mean 1.50 cubic yards and standard deviation 0.25 cubic yards. During the same week, coal is removed from the storage bin and loaded into 4 railroad cars. The amount of coal loaded into each railroad car is normally distributed with mean 7.25 cubic yards and standard deviation 0.50 cubic yards. The amounts added to the storage bin or removed from the storage bin are mutually independent.

Calculate the probability that the storage bin contains more coal at the end of the week than it had at the beginning of the week.

• 0.56
• 0.63
• 0.67
• 0.75
• 0.98

Under a liability insurance policy, losses are uniformly distributed on [math][0, b][/math], where [math]b[/math] is a positive constant. There is a deductible of [math]b/2[/math]. Calculate the ratio of the variance of the claim payment (greater than or equal to zero) from a given loss to the variance of the loss.

• 1:8
• 3:16
• 1:4
• 5:16
• 1:2

Let [math]X[/math] be the percentage score on a college-entrance exam for students who did not participate in an exam-preparation seminar. [math]X[/math] is modeled by a uniform distribution on [a, 100].

Let [math]Y[/math] be the percentage score on a college-entrance exam for students who did participate in an exam-preparation seminar. [math]Y[/math] is modeled by a uniform distribution on [1.25a, 100].

It is given that [math]\operatorname{E}( X^2 ) = 19, 600. [/math]

Calculate the 80th percentile of [math]Y[/math].

• 0.64
• 0.74
• 0.85
• 0.87
• 0.94

Bridies' Bearing Works manufactures bearing shafts whose diameters are normally distributed with parameters [math]\mu = 1[/math], [math]\sigma = .002[/math]. The buyer's specifications require these diameters to be [math]1.000 \pm .003[/math] cm. If the manufacturer improves her quality control, she can reduce the value of [math]\sigma[/math]. Find the greatest value of [math]\sigma[/math] that will ensure that no more than 1 percent of her shafts are likely to be rejected?

• 12.9 * 10^-3
• 13.28 * 10 ^-3
• 13.6 * 10 ^-3
• 14 * 10 ^-3
• 14.28 * 10 ^-3

References

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

Losses have a uniform distribution on [0,b]. When a deductible equal to r*b is in effect, the payment variance is 0.75(1-r)² times the payment variance without deductible. Determine which interval contains the value of r.

• [0.55,0.6]
• [0.68,0.73]
• [0.75,0.79]
• [0.82,0.85]
• [0.9,0.95]

A policy's loss is uniform on [0,2000] with a deductible of $500 and limit, applied before the deductible, of $1,500. Determine the expected payment, given that the payment is non-zero.

• 501.55
• 625.67
• 666.67
• 700.67
• 872.55

A driver and a passenger are in a car accident. Each of them independently has probability 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0, 1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1.

• 0.510
• 0.534
• 0.600
• 0.628
• 0.800

Losses, [math]X[/math], under an insurance policy are exponentially distributed with mean 10. For each loss, the claim payment [math]Y[/math] is equal to the amount of the loss in excess of a deductible [math]d \gt 0 [/math].

Calculate [math]\operatorname{Var}(Y)[/math].

• [math]100-d[/math]
• [math](10 − d)^2[/math]
• [math]100e^{-d/10}[/math]
• [math]100(2e^{-d/100}-e^{-d/5})[/math]
• [math](10-d)^2(2e^{-d/100}-e^{-d/5})[/math]

The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%.

Calculate the percentage reduction on the variance of the claim payment.

• 1%
• 5%
• 10%
• 20%
• 25%

A government employee’s yearly dental expense follows a uniform distribution on the interval from 200 to 1200. The government’s primary dental plan reimburses an employee for up to 400 of dental expense incurred in a year, while a supplemental plan pays up to 500 of any remaining dental expense. Let [math]Y[/math] represent the yearly benefit paid by the supplemental plan to a government employee.

Calculate [math]\operatorname{Var}(Y)[/math].

• 20,833
• 26,042
• 41,042
• 53,333
• 83,333