You are given the following:

• Claim frequency and claim size are independent
• Monthly claim frequency is Poisson distributed with mean 3
• The claim size distribution is uniform on [0,1000]

If [math]S[/math] is the annual loss, determine the variance of [math]S[/math]

• 8,000,000
• 10,500,000
• 11,000,000
• 12,000,000
• 13,000,000

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

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

A fair die is rolled repeatedly. Let [math]X[/math] be the number of rolls needed to obtain a 5 and [math]Y[/math] the number of rolls needed to obtain a 6.

Calculate [math]\operatorname{E}(X | Y = 2). [/math]

• 5.0
• 5.2
• 6.0
• 6.6
• 6.8

New dental and medical plan options will be offered to state employees next year. An actuary uses the following density function to model the joint distribution of the proportion [math]X[/math] of state employees who will choose Dental Option 1 and the proportion [math]Y[/math] who will choose Medical Option 1 under the new plan options:

Calculate [math]\operatorname{Var} (Y | X = 0.75)[/math].

• 0.000
• 0.061
• 0.076
• 0.083
• 0.141

A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy. To purchase the supplemental policy, an employee must first purchase the basic policy. Let [math]X[/math] denote the proportion of employees who purchase the basic policy, and [math]Y[/math] the proportion of employees who purchase the supplemental policy. Let [math]X[/math] and [math]Y[/math] have the joint density function [math]f(x,y) = 2(x+y)[/math] on the region where the density is positive.

Given that 10% of the employees buy the basic policy, calculate the probability that fewer than 5% buy the supplemental policy.

• 0.010
• 0.013
• 0.108
• 0.417
• 0.500

Let [math]N[/math] denote the number of accidents occurring during one month on the northbound side of a highway and let [math]S[/math] denote the number occurring on the southbound side. Suppose that [math]N[/math] and [math]S[/math] are jointly distributed as indicated in the table.

Calculate [math]\operatorname{Var}(N | N + S = 2) [/math].

• 0.48
• 0.55
• 0.67
• 0.91
• 1.25

A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease present and 0 for disease not present. Let [math]X[/math] denote the disease state (0 or 1) of a patient, and let [math]Y[/math] denote the outcome of the diagnostic test. The joint probability function of [math]X[/math] and [math]Y[/math] is given by:

Calculate [math]\operatorname{Var}(Y | X = 1) .[/math]

• 0.13
• 0.15
• 0.20
• 0.51
• 0.71