Suppose the claim frequency [math]N[/math] has a probability function defined recursively by

where [math]a\gt0[/math] and [math]k \geq 1 [/math]. Which of the following expressions equals the expected value of [math]N[/math]?

1. [math]a/(1-a)[/math]
2. [math]a [/math]
3. [math]a/(1-a^2)[/math]
4. [math]a^2/(1-a^2) [/math]
5. [math]1-a [/math]

Suppose [math]F(x)[/math] is a continuous cumulative probability distribution function with [math]\lim_{x\rightarrow 1}F(x)=1[/math] and [math]F(x)\gt0[/math] for all [math]x[/math]. For which of the following [math]g(x)[/math] is [math]F(g(x))[/math] also a cumulative probability distribution function?

• [math]x^2[/math]
• [math]\sqrt{|x| + 1} [/math]
• [math]e^{-x}[/math]
• [math](1 + e^{-x})^{-1}[/math]
• [math]1-\ln(1 + e^{-x})[/math]

Suppose the loss has a continuous cumulative distribution function [math]F(x)[/math] with the following values:

If [math]P[/math] denotes the payment, determine the 25^th percentile of [math]P^{-1}[/math].

• 800
• 1/800
• 250
• 1/250
• 1/1000

The number of injury claims per month is modeled by a random variable [math]N[/math] with

, for nonnegative integers, [math]n[/math]. Calculate the probability of at least one claim during a particular month, given that there have been at most four claims during that month.

• 1/3
• 2/5
• 1/2
• 3/5
• 5/6

The loss due to a fire in a commercial building is modeled by a random variable [math]X[/math] with density function

Given that a fire loss exceeds 8, calculate the probability that it exceeds 16.

• 1/25
• 1/9
• 1/8
• 1/3
• 3/7

The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function [math]f(x)[/math], where [math]f(x)[/math] is proportional to [math](10+x)^{-2}[/math] on the interval.

Calculate the probability that the lifetime of the machine part is less than 6.

• 0.04
• 0.15
• 0.47
• 0.53
• 0.94

A group insurance policy covers the medical claims of the employees of a small company. The value, [math]V[/math], of the claims made in one year is described by [math]V = 100,000Y[/math] where [math]Y[/math] is a random variable with density function

where [math]k[/math] is a constant. Calculate the conditional probability that [math]V[/math] exceeds 40,000, given that [math]V[/math] exceeds 10,000.

• 0.08
• 0.13
• 0.17
• 0.20
• 0.51

An insurance policy pays for a random loss [math]X[/math] subject to a deductible of [math]C[/math], where [math]0 \lt C \lt 1[/math] . The loss amount is modeled as a continuous random variable with density function

Given a random loss [math]X[/math], the probability that the insurance payment is less than 0.5 is equal to 0.64. Calculate [math]C[/math].

• 0.1
• 0.3
• 0.4
• 0.6
• 0.8

An insurance policy pays 100 per day for up to three days of hospitalization and 50 per day for each day of hospitalization thereafter. The number of days of hospitalization, [math]X[/math], is a discrete random variable with probability function

Determine the expected payment for hospitalization under this policy.

• 123
• 210
• 220
• 270
• 367

Let [math]X[/math] be a continuous random variable with density function

Calculate the expected value of [math]X[/math].

• 1/5
• 3/5
• 1
• 28/15
• 12/5