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 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

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

A manufacturer’s annual losses follow a distribution with density function

To cover its losses, the manufacturer purchases an insurance policy with an annual deductible of 2. Calculate the mean of the manufacturer’s annual losses not paid by the insurance policy.

• 0.84
• 0.88
• 0.93
• 0.95
• 1.00

An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, [math]Y[/math], follows a distribution with density function:

Calculate the expected value of the benefit paid under the insurance policy.

• 1.0
• 1.3
• 1.8
• 1.9
• 2.0

An insurance company’s monthly claims are modeled by a continuous, positive random variable [math]X[/math], whose probability density function is proportional to [math](1 + x)^{- 4}[/math] for [math] x \gt 0 [/math] .

Calculate the company’s expected monthly claims.

• 1/6
• 1/3
• 1/2
• 1
• 3

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

Calculate the value of [math]p[/math] such that [math]\operatorname{E}(X) = 2 [/math].

• 1
• 2.5
• 3
• 5
• There is no such [math]p[/math]

The lifetime of a light bulb has density function, [math]f[/math], where [math]f(x)[/math] is proportional to

Calculate the mode of this distribution.

• 0.00
• 0.79
• 1.26
• 4.42
• 5.00

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

Calculate [math]\operatorname{P}[ X \leq 0.5 | X \leq 1.0].[/math]

• 0.433
• 0.547
• 0.632
• 0.731
• 0.865

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

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

• 800
• 1/800
• 250
• 1/250
• 1/1000