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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 \(\left(x_{1}\right)\) and to repair a computer \(\left(x_{2}\right) .\) 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 \(W\).

Step by step solution

01

Understand the Problem

We have two random variables: \(x_1\) for examining the computer and \(x_2\) for repairing it. They are independent, which means the combined random variable \(W = x_1 + x_2\) summing their times.

02

Find the Mean of W

For two independent random variables, the mean of their sum is the sum of their means. Hence, \(\mu_W = \mu_1 + \mu_2\). Calculate \(\mu_W = 28.1 + 90.5 = 118.6\) minutes.

03

Find the Variance of W

The variance of the sum of independent random variables is the sum of their variances. Thus, \(\sigma^2_W = \sigma^2_1 + \sigma^2_2\). Compute \(\sigma^2_W = 8.2^2 + 15.2^2 = 67.24 + 231.04 = 298.28\) minutes squared.

04

Calculate the Standard Deviation of W

The standard deviation is the square root of the variance. Thus, \(\sigma_W = \sqrt{\sigma^2_W} = \sqrt{298.28} \approx 17.27\) minutes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean of Random Variables

When dealing with independent random variables such as \(x_1\) and \(x_2\), finding the mean of their sum is quite straightforward.
The core idea is that the average (mean) of a combined random variable \(W = x_1 + x_2\) is just the sum of their individual averages.
This is expressed mathematically as:

• \(\mu_W = \mu_1 + \mu_2\)

Let's look at the computer repair shop example. We have:

• \(\mu_1 = 28.1\) minutes for examination
• \(\mu_2 = 90.5\) minutes for repair

To find the total average time \(\mu_W\):

• Calculate \(\mu_W = 28.1 + 90.5 = 118.6\) minutes.

The calculation is intuitive, showing the simplicity and ease in combining means of independent variables. It allows efficient planning for the tasks at the repair shop since the mean of their total time is easily computed.

Variance Calculation

Variance helps us understand how much variability there is in time from the mean.
For the sum of independent random variables like \(x_1\) and \(x_2\), the variance of the sum \(W\) is straightforward to calculate as well.
Variance is always additive for independent variables, meaning:

• \(\sigma^2_W = \sigma^2_1 + \sigma^2_2\)

From the given exercise, we have:

• \(\sigma_1 = 8.2\) minutes, so \(\sigma^2_1 = 8.2^2 = 67.24\)
• \(\sigma_2 = 15.2\) minutes, so \(\sigma^2_2 = 15.2^2 = 231.04\)

Now, let's find the total variance \(\sigma^2_W\):

• Calculate \(\sigma^2_W = 67.24 + 231.04 = 298.28\) minutes squared.

This step is crucial because variance provides insight into how unpredictably the examination and repair times may fluctuate. The result is more variance when the total process time is considered.

Standard Deviation

Standard deviation offers a more direct interpretation of variability by returning to the original units of measure, in this case, minutes.
Calculating the standard deviation for the combined random variable \(W\) involves taking the square root of the total variance found in the previous step.

• \(\sigma_W = \sqrt{\sigma^2_W}\)

From the variance calculation:

• \(\sigma^2_W = 298.28\) minutes squared

Taking the square root to find the standard deviation:

• \(\sigma_W = \sqrt{298.28} \approx 17.27\) minutes.

Standard deviation tells us how much the time to complete both tasks is expected to vary around the mean time of 118.6 minutes.
It helps to give a reasonable estimation of the time range when booking appointments or planning repairs, making it practical in real-world scenarios.