Losses are assumed to have an exponential distribution with unknown mean [math]\theta[/math]. Ten policies have a deductible of $500 and the total number of payments for the ten policies has the moment generating function

Determine the mean [math]\theta[/math].

• [100, 150]
• [300, 320]
• [475, 525]
• [1750, 2000]
• [2200, 2400]

The logarithm of claim severity for a risk is normally distributed with mean 2 and standard deviation 1. Determine the variance of claim severity for the risk.

• e
• 255
• 391
• 403
• 1,042

A portfolio of policies has the following properties:

• Claim frequency is Poisson distributed with unknown mean [math]\lambda [/math] varying from policy to policy.
• The mean [math]\lambda [/math] is distributed among the risks in the portfolio according to a gamma distribution.
• The expected claim frequency for a randomly selected policy is 2.
• The probability of a claim frequency observation equaling zero for a randomly selected risk is 27/125.

Determine the variance of [math]\lambda [/math].

• 1
• 4/3
• 2
• 3
• 27/4

An actuary determines that the claim size for a certain class of accidents is a random variable, [math]X[/math], with moment generating function

Calculate the standard deviation of the claim size for this class of accidents.

• 1,340
• 5,000
• 8,660
• 10,000
• 11,180

A company insures homes in three cities, J, K, and L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are mutually independent. The moment generating functions for the loss distributions of the cities are:

Let [math]X[/math] represent the combined losses from the three cities. Calculate [math]\operatorname{E}[X^3][/math] .

• 1,320
• 2,082
• 5,760
• 8,000
• 10,560

[math]X[/math] and [math]Y[/math] are independent random variables with common moment generating function [math]M(t) = \exp(t^2/ 2).[/math] Let [math]W = X + Y[/math] and [math]Z = Y - X[/math]. Determine the joint moment generating function, [math]M(t_1,t_2)[/math] of [math]W[/math] and [math]Z[/math].

• [math]\exp(2t_1^2 + 2t_2^2 )[/math]
• [math]\exp[(t_1 − t_2 )^2 ][/math]
• [math]\exp[(t_1 + t_2 )^2 ][/math]
• [math]\exp(2t_1t_2 )[/math]
• [math]\exp(t_1^2 + t_2^2 )[/math]

Let [math]X_1, X_2, X_3[/math] be a random sample from a discrete distribution with probability function

Determine the moment generating function, [math]M(t)[/math], of [math]Y = X_1X_2X_3 [/math].

• [math]\frac{19}{27} + \frac{8}{27}e^t[/math]
• [math]1 + 2e^t[/math]
• [math](\frac{1}{3} + \frac{2}{3}e^t)^3[/math]
• [math]\frac{1}{27} + \frac{8}{27}e^{3t}[/math]
• [math]\frac{1}{3} + \frac{2}{3}e^{3t}[/math]

The value of a piece of factory equipment after three years of use is 100(0.5)[math]X[/math] where [math]X[/math] is a random variable having moment generating function

Calculate the expected value of this piece of equipment after three years of use.

• 12.5
• 25.0
• 41.9
• 70.7
• 83.8