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 depends on parameters [math]a,b[/math]: when [math]a \lt b [/math] the distribution is uniform on [math][a,b][/math] and when [math]a = b [/math] the claim size equals [math]b [/math] with probability 1.

If [math]S[/math] is the annual loss, determine the maximum of [math]\operatorname{E}[S^2]/\operatorname{E}[S]^2[/math] over all possible values a,b.

• 1/2
• 3/4
• 1
• 4/3
• 5/3

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]

The intensity of a hurricane is a random variable that is uniformly distributed on the interval [0, 3]. The damage from a hurricane with a given intensity [math]y[/math] is exponentially distributed with a mean equal to [math]y[/math]. Calculate the variance of the damage from a random hurricane.

• 1.73
• 1.94
• 3.00
• 3.75
• 6.00

An auto insurance policy will pay for damage to both the policyholder’s car and the other driver’s car in the event that the policyholder is responsible for an accident. The size of the payment for damage to the policyholder’s car, [math]X[/math], has a marginal density function of 1 for [math] 0 \lt x \lt 1 [/math]. Given [math]X=x[/math], the size of the payment for damage to the other driver’s car, [math]Y[/math], has conditional density of 1 for [math]x \lt y \lt x + 1 [/math].

Given that the policyholder is responsible for an accident, calculate the probability that the payment for damage to the other driver’s car will be greater than 0.5.

• 3/8
• 1/2
• 3/4
• 7/8
• 15/16

Let [math]N_1[/math] and [math]N_2[/math] represent the numbers of claims submitted to a life insurance company in April and May, respectively. The joint probability function of [math]N_1[/math] and [math]N_2[/math] is

Calculate the expected number of claims that will be submitted to the company in May, given that exactly 2 claims were submitted in April.

• [math]\frac{3}{16}(e^2 -1)[/math]
• [math]\frac{3}{16}e^2[/math]
• [math]\frac{3e}{4-e}[/math]
• [math]e^2 -1 [/math]
• [math]e^2[/math]

A motorist makes three driving errors, each independently resulting in an accident with probability 0.25. Each accident results in a loss that is exponentially distributed with mean 0.80. Losses are mutually independent and independent of the number of accidents. The motorist’s insurer reimburses 70% of each loss due to an accident.

Calculate the variance of the total unreimbursed loss the motorist experiences due to accidents resulting from these driving errors.

• 0.0432
• 0.0756
• 0.1782
• 0.2520
• 0.4116

Automobile policies are separated into two groups: low-risk and high-risk. Actuary Rahul examines low-risk policies, continuing until a policy with a claim is found and then stopping. Actuary Toby follows the same procedure with high-risk policies. Each low-risk policy has a 10% probability of having a claim. Each high-risk policy has a 20% probability of having a claim. The claim statuses of polices are mutually independent.

Calculate the probability that Actuary Rahul examines fewer policies than Actuary Toby

• 0.2857
• 0.3214
• 0.3333
• 0.3571
• 0.4000