Choose a number [math]U[/math] from the interval [math][0,1][/math] with uniform distribution. Find the density for the random variables [math]Y = |U - 1/2|[/math].

• [[math]]f(y) = \begin{cases}2|y-1/2|, \, 0 \leq y \leq 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}1/2, \, 0 \leq y \leq 2\\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}2 - |y-1/2|, \, 0 \leq y \leq 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}2, \, 0 \leq y \leq 1/2 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]
• [[math]]f(y) = \begin{cases}2y, \, 0 \leq y \leq 1 \\ 0, \, \textrm{Otherwise} \end{cases}[[/math]]

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

A number [math]U[/math] is chosen at random in the interval [math][0,1][/math]. Find the probability that [math]T = U/(1 - U) \lt 1/4[/math].

• 1/6
• 1/5
• 1/4
• 1/2
• 2/3

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Let [math]T[/math] denote the time in minutes for a customer service representative to respond to 10 telephone inquiries. [math]T[/math] is uniformly distributed on the interval with endpoints 8 minutes and 12 minutes. Let [math]R[/math] denote the average rate, in customers per minute, at which the representative responds to inquiries, and let [math]f(r)[/math] be the density function for [math]R[/math].

Determine [math]f(r)[/math], for [math]\frac{10}{12} \lt r \lt \frac{10}{8}[/math].

• 12/5
• 3-5/2r
• [math]3r-5\ln(r)/2[/math]
• [math]\frac{10}{r^2}[/math]
• [math]\frac{5}{2r^2}[/math]

An actuary models the lifetime of a device using the random variable [math]Y = 10X^{0.8}[/math], where [math]X[/math] is an exponential random variable with mean 1. Let [math]f(y)[/math] be the density function for [math]Y[/math].

Determine [math]f(y)[/math] for [math]y\gt0[/math].

• [math]10 y^{0.8} \exp(−8 y^{−0.2} )[/math]
• [math]8 y^{−0.2} \exp(−10 y^{0.8} )[/math]
• [math]8 y^{−0.2} \exp[−(0.1y)^{1.25} ][/math]
• [math](0.1y )^{1.25} \exp[−0.125(0.1y)^{0.25} ][/math]
• [math]0.125(0.1y )^{0.25} \exp[−(0.1y )^{1.25} ][/math]

The monthly profit of Company I can be modeled by a continuous random variable with density function [math]f[/math]. Company II has a monthly profit that is twice that of Company I. Let [math]g[/math] be the density function for the distribution of the monthly profit of Company II.

Determine [math]g(y)[/math] where it is not zero.

• [math]\frac{1}{2}f(\frac{y}{2})[/math]
• [math]f(\frac{y}{2})[/math]
• [math]2f(\frac{y}{2})[/math]
• [math]2f(y)[/math]
• [math]2f(2y)[/math]

An investment account earns an annual interest rate [math]R[/math] that follows a uniform distribution on the interval (0.04, 0.08). The value of a 10,000 initial investment in this account after one year is given by [math]V = 10 000e^R. [/math] Let [math]F[/math] be the cumulative distribution function of [math]V[/math]. Determine [math]F(v)[/math] for values of [math]v[/math] that satisfy [math]0 \lt F(v) \lt 1 [/math].

• [math]\frac{10000e^{v /10000} − 10408}{425}[/math]
• [math]25e^{v/10000} - 0.04[/math]
• [math]\frac{v-10408}{10833-10408}[/math]
• [math]\frac{25}{v}[/math]
• [math]25\left[ \ln(v/10000) - 0.04 \right][/math]

The time, [math]T[/math], that a manufacturing system is out of operation has cumulative distribution function

The resulting cost to the company is [math]Y = T^2[/math]. Let [math]g[/math] be the density function for [math]Y[/math]. Determine [math]g(y)[/math], for [math]y \gt 4 [/math].

• [math]\frac{4}{y^2}[/math]
• [math]\frac{8}{y^{3/2}}[/math]
• [math]\frac{8}{y^3}[/math]
• [math]\frac{16}{y}[/math]
• [math]\frac{1024}{y^5}[/math]

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

Suppose that [math]Y = \exp(\exp(X)) [/math], determine the mode of the distribution of [math]Y[/math].

• 2.7
• 3.9
• 5.2
• 6.4
• 7.8

Losses for year 1 equal

with [math]X[/math] a non-negative random variable bounded by 1 with cumulative distribution function

Determine the probability that losses exceed $300.

• 0.064
• 0.0878
• 0.148
• 0.296
• 0.444

You are given the following:

• Year 1 loss has an exponential distribution with mean $1,250.
• Claim costs rise 5% from year 1 to year 2
• The policy has a deductible of $400 in effect during years 1 and 2.
• x₁ denotes the probability that payment exceeds $500 in year 1, given that payment is positive.
• x₂ denotes the probability that payment exceeds $500 in year 2, given that payment is positive.

Which interval contains the ratio x₂/x₁?

• [0.5, 0.7]
• [0.98, 1.03]
• [1.1, 1.2]
• [1.3, 1.4]
• 1.5+