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

An insurance policy is written to cover a loss [math]X[/math] where [math]X[/math] has density function

The time (in hours) to process a claim of size [math]x[/math], where [math]0\lt x \lt 2[/math] , is uniformly distributed on the interval from [math]x[/math] to [math]2x[/math].

Calculate the probability that a randomly chosen claim on this policy is processed in three hours or more.

• 0.17
• 0.25
• 0.32
• 0.58
• 0.83