Find [math]E(X^Y)[/math], where [math]X[/math] and [math]Y[/math] are independent random variables which are uniform on [math][0, 1][/math].

• 0.6931
• 0.7131
• 0.7344
• 0.7544
• 0.775

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|)[/math].

• 1/5
• 1/3
• 1/2
• 2/3
• 4/5

References

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

The number of hurricanes that will hit a certain house in the next ten years is Poisson distributed with mean 4. Each hurricane results in a loss that is exponentially distributed with mean 1000. Losses are mutually independent and independent of the number of hurricanes.

Calculate the variance of the total loss due to hurricanes hitting this house in the next ten years.

• 4,000,000
• 4,004,000
• 8,000,000
• 16,000,000
• 20,000,000

Losses follow an exponential distribution with mean 1. Two independent losses are observed.

Calculate the probability that either of the losses is more than twice the other.

• 1/6
• 1/4
• 1/3
• 1/2
• 2/3

An actuary is studying hurricane models. A year is classified as a high, medium, or low hurricane year with probabilities 0.1, 0.3, and 0.6, respectively. The numbers of hurricanes in high, medium, and low years follow Poisson distributions with means 20, 15, and 10, respectively.

Calculate the variance of the number of hurricanes in a randomly selected year.

• 11.25
• 12.50
• 12.94
• 13.42
• 23.75

On Main Street, a driver’s speed just before an accident is uniformly distributed on [5, 20]. Given the speed, the resulting loss from the accident is exponentially distributed with mean equal to three times the speed.

Calculate the variance of a loss due to an accident on Main Street.

• 525
• 1463
• 1575
• 1632
• 1744

For a certain insurance company, 10% of its policies are Type A, 50% are Type B, and 40% are Type C. The annual number of claims for an individual Type A, Type B, and Type C policy follow Poisson distributions with respective means 1, 2, and 10.

Let [math]X[/math] represent the annual number of claims of a randomly selected policy. Calculate the variance of [math]X[/math].

• 5.10
• 16.09
• 21.19
• 42.10
• 47.20

Individuals purchase both collision and liability insurance on their automobiles. The value of the insured’s automobile is V. Assume the loss L on an automobile claim is a random variable with cumulative distribution function

Calculate the probability that the loss on a randomly selected claim is greater than the value of the automobile.

• 0.00
• 0.10
• 0.25
• 0.75
• 0.90

An actuary has done an analysis of all policies that cover two cars. 70% of the policies are of type A for both cars, and 30% of the policies are of type B for both cars. The number of claims on different cars across all policies are mutually independent. The distributions of the number of claims on a car are given in the following table.

Calculate the probability that exactly one of the four policies has the same number of claims on both covered cars.

• 0.104
• 0.250
• 0.285
• 0.417
• 0.739

In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected.

Calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer.

• 0.403
• 0.500
• 0.547
• 0.600
• 0.625