Salary An employer pays a mean salary for a 5-day workweek of 250 with a standard deviation of 2

step 1 Let X pay the salary expenses during five -day work week. Let Y be the salary expenses during we

Salary An employer pays a mean salary for a 5-day workweek...

Salary An employer pays a mean salary for a 5-day workweek of $\$1250$ with a standard deviation of $\$129.$ On the weekends, his salary expenses have a mean of $450 with a standard deviation of $\$57.$ What is the mean and standard deviation of his total weekly salaries?

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Key Concepts

Expected Value (Mean)

The mean, or expected value, of a random variable represents its long?run average outcome over many trials. In problems involving sums of random variables, the linearity of expectation allows one to simply add the means of the individual components to get the overall mean, regardless of whether the variables are independent.

Variance and Standard Deviation

The standard deviation is a measure of the dispersion or spread of a distribution around its mean, and it is the square root of the variance. When combining independent random variables, variances add up, so one can find the overall variability of the sum by summing the individual variances and then taking the square root to determine the total standard deviation.

Linear Combination of Independent Random Variables

When dealing with a linear combination (such as the sum) of independent random variables, the overall expected value is the sum of the individual expected values, and the overall variance is the sum of the individual variances. This principle is essential when calculating the total weekly salary from separate salary components that are assumed to be independent.

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