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

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

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

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

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

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.

A loss variable [math]L[/math] has a density function that is proportional to

The parameters [math]\theta [/math] and [math]\alpha [/math] are random with the following joint distribution

Determine the standard deviation of [math]L[/math] to the nearest integer.

• 298
• 1,088
• 1,220
• 1,279
• 1,565

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

for a constant [math]c[/math]. Determine the marginal density of [math]2Y^{1/2}[/math] given [math]X=1/2[/math].

• [[math]] g(z)= \begin{cases} \frac{z^7}{6}, \sqrt{2} \lt z \lt 2^{3/4} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} \frac{64z^3}{3}, \frac{1}{2} \lt z \lt \frac{1}{\sqrt{2}} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} z^3, \sqrt{2} \lt z \lt 2^{3/4} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} \frac{255z^7}{1688}, \frac{1}{2} \lt z \lt \frac{1}{\sqrt{2}} \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]
• [[math]] g(z)= \begin{cases} \frac{2^{7/2}z^{5/2}}{5}, 0 \lt z \lt 2 \\ 0, \, \textrm{Otherwise} \end{cases} [[/math]]

The loss in year 1, [math]X[/math], has probability density function

A deductible equalling the loss in year 1 is applicable in year 2. If the payment in year 2 equals [math]Y[/math], determine the joint density function for [math]X,Y[/math], given that [math]Y\gt0[/math].

• [[math]] \begin{align*} g(x,y) &= \begin{cases} \frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
• [[math]] \begin{align*} g(x,y) &= \begin{cases} \frac{\alpha^2}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
• [[math]] \begin{align*} g(x,y) &= \begin{cases} \frac{\alpha^2\theta^{\alpha}}{(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
• [[math]] \begin{align*} g(x,y) &= \begin{cases} \frac{\alpha^2\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]
• [[math]] \begin{align*} g(x,y) &= \begin{cases} \frac{\alpha\theta^{\alpha}}{(x+\theta)(y + x + \theta)^{\alpha + 1}}, y \gt 0, x \gt 0 \\ 0, \, \textrm{Otherwise} \end{cases} \\ \end{align*} [[/math]]

You are given the following about a portfolio of risks:

• Risks are classified into three classes: 10% belong to class A, 30% belong to class B and 60% belong to class C.
• Losses for each risk are uniformly distributed on an interval [a,b] with a and b dependent on class:

Determine the expected loss for a randomly selected risk given that the loss is greater than 1,000.

• 1,065
• 1,238
• 1,313
• 1,587
• 1,597