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Combination of Random Variables: Repair Service A computer

Chapter , Problem 16

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QUESTION:

Combination of Random Variables: Repair Service A computer repair shop has two work centers. The first center examines the computer to see what is wrong, and the second center repairs the computer. Let \(x_{1}\) and \(x_{2}\) be random variables representing the lengths of time in minutes to examine a computer (\(x_{1}\)) and to repair a computer (\(x_{2}\)). Assume \(x_{1}\) and \(x_{2}\) are independent random variables. Long-term history has shown the following times:

Examine computer, \(x_{1}: \mu_{1}=28.1\) minutes; \(\sigma_{1}=8.2\) minutes

Repair computer, \(x_{2}: \mu_{2}=90.5\) minutes; \(\sigma_{2}=15.2\) minutes

(a) Let \(W=x_{1}+x_{2}\) be a random variable representing the total time to examine and repair the computer. Compute the mean, variance, and standard deviation of W.

(b) Suppose it costs $1.50 per minute to examine the computer and $2.75 per minute to repair the computer. Then \(W=1.50 x_{1}+2.75 x_{2}\) is a random variable representing the service charges (without parts). Compute the mean, variance, and standard deviation of W.

(c) The shop charges a flat rate of $1.50 per minute to examine the computer, and if no repairs are ordered, there is also an additional $50 service charge. Let \(L=1.5 x_{1}+50\). Compute the mean, variance, and standard deviation of L.

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QUESTION:

Combination of Random Variables: Repair Service A computer repair shop has two work centers. The first center examines the computer to see what is wrong, and the second center repairs the computer. Let \(x_{1}\) and \(x_{2}\) be random variables representing the lengths of time in minutes to examine a computer (\(x_{1}\)) and to repair a computer (\(x_{2}\)). Assume \(x_{1}\) and \(x_{2}\) are independent random variables. Long-term history has shown the following times:

Examine computer, \(x_{1}: \mu_{1}=28.1\) minutes; \(\sigma_{1}=8.2\) minutes

Repair computer, \(x_{2}: \mu_{2}=90.5\) minutes; \(\sigma_{2}=15.2\) minutes

(a) Let \(W=x_{1}+x_{2}\) be a random variable representing the total time to examine and repair the computer. Compute the mean, variance, and standard deviation of W.

(b) Suppose it costs $1.50 per minute to examine the computer and $2.75 per minute to repair the computer. Then \(W=1.50 x_{1}+2.75 x_{2}\) is a random variable representing the service charges (without parts). Compute the mean, variance, and standard deviation of W.

(c) The shop charges a flat rate of $1.50 per minute to examine the computer, and if no repairs are ordered, there is also an additional $50 service charge. Let \(L=1.5 x_{1}+50\). Compute the mean, variance, and standard deviation of L.

ANSWER:

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Mean, variance, and standard deviation are three important statistical measures that are used to describe a dataset. They are all related to each other, but they measure different things. They provide insights into the central tendency and the spread or dispersion of the data points within the dataset.

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