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 loss in year 2, with deductible in effect, equals [math]Y[/math], determine the joint density function for [math]X,Y[/math].

• [[math]] \begin{align*} f_{X,Y}(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*} f_{X,Y}(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*} f_{X,Y}(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*} f_{X,Y}(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*} f_{X,Y}(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

You are given the following about a risk:

• Claim frequency and claim severity are independent
• Monthly claim frequency is Poisson distributed
• Claim severity is uniformly distributed.

If [math]S[/math] is an annual loss for the risk, determine the maximum of [math]\operatorname{E}[S^2]/\operatorname{E}[S]^2[/math].

• 1/2
• 3/4
• 1
• 4/3
• 5/3

Let [math]X[/math] and [math]Y[/math] be continuous random variables with joint density function

Calculate [math] \operatorname{P}[Y \lt X | X = 1/3][/math]

• 1/27
• 2/27
• 1/4
• 1/3
• 4/9

Once a fire is reported to a fire insurance company, the company makes an initial estimate, [math]X[/math], of the amount it will pay to the claimant for the fire loss. When the claim is finally settled, the company pays an amount, [math]Y[/math], to the claimant. The company has determined that [math]X[/math] and [math]Y[/math] have the joint density function

Given that the initial claim estimated by the company is 2, calculate the probability that the final settlement amount is between 1 and 3.

• 1/9
• 2/9
• 1/3
• 2/3
• 8/9

The stock prices of two companies at the end of any given year are modeled with random variables [math]X[/math] and [math]Y[/math] that follow a distribution with joint density function

Determine the conditional variance of [math]Y[/math] given that [math]X = x[/math].

• 1/12
• 7/6
• [math]x + 1/2 [/math]
• [math]x^2 - 1/6[/math]
• [math]x^2 +x + 1/3[/math]

An actuary determines that the annual number of tornadoes in counties P and Q are jointly distributed as follows:

Calculate the conditional variance of the annual number of tornadoes in county Q, given that there are no tornadoes in county P.

• 0.51
• 0.84
• 0.88
• 0.99
• 1.76

You are given the following information about [math]N[/math], the annual number of claims for a randomly selected insured:

Let [math]S[/math] denote the total annual claim amount for an insured. When [math]N = 1 [/math], [math]S[/math] is exponentially distributed with mean 5. When [math]N \gt 1 [/math], [math]S[/math] is exponentially distributed with mean 8.

Calculate [math]\operatorname{P}(4 \lt S \lt 8) [/math]

• 0.04
• 0.08
• 0.12
• 0.24
• 0.25