A device containing two key components fails when, and only when, both components fail. The lifetimes, [math]T_1 [/math] and [math]T_2[/math] of these components are independent with common density function

The cost, [math]X[/math], of operating the device until failure is [math]2T_1 + T_2[/math] . Let [math]g[/math] be the density function for [math]X[/math].

Determine [math]g(x)[/math], for [math]x \gt 0 [/math].

• [math]e^{−x/2} − e^{−x}[/math]
• [math]2(e^{-x/2} - e^{-x} )[/math]
• [math]\frac{x^2e^{-x}}{2}[/math]
• [math]\frac{e^{-x/2}}{2}[/math]
• [math]\frac{e^{-x/3}}{3}[/math]

Two instruments are used to measure the height, [math]h[/math], of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation 0.0056 [math]h [/math]. The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation 0.0044 [math]h[/math]. The errors from the two instruments are independent of each other.

Calculate the probability that the average value of the two measurements is within 0.005 [math]h[/math] of the height of the tower.

• 0.38
• 0.47
• 0.68
• 0.84
• 0.90

A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have mutually independent lifetimes.

Calculate the smallest number of bulbs to be purchased so that the succession of light bulbs produces light for at least 40 months with probability at least 0.9772.

• 14
• 16
• 20
• 40
• 55

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 company offers earthquake insurance. Annual premiums are modeled by an exponential random variable with mean 2. Annual claims are modeled by an exponential random variable with mean 1. Premiums and claims are independent. Let [math]X[/math] denote the ratio of claims to premiums, and let [math]f[/math] be the density function of [math]X[/math].

Determine [math]f(x)[/math], where it is positive.

• [math]\frac{1}{2x+1}[/math]
• [math]\frac{2}{(2x+1)^2}[/math]
• [math]e^{-x}[/math]
• [math]2e^{-2x}[/math]
• [math]xe^{-x}[/math]

For Company A there is a 60% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 10,000 and standard deviation 2,000.

For Company B there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 9,000 and standard deviation 2,000.

The total claim amounts of the two companies are independent.

Calculate the probability that, in the coming year, Company B’s total claim amount will exceed Company A’s total claim amount.

• 0.180
• 0.185
• 0.217
• 0.223
• 0.240