A charitable organization solicits donations by telephone....

A charitable organization solicits donations by telephone. Employees are paid $$\$ 60$$ plus $20 \%$ of the money their calls generate each week. The amount of money generated in a week can be viewed as a random variable with a mean of $$\$ 700$$ and a standard deviation of $$\$ 130$$. Find the mean and standard deviation of an $\mathrm{em}$ ployee's total pay in a week.

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Key Concepts

Linear Transformations of Random Variables

This concept involves modifying a random variable by applying a scaling factor and/or an additive constant. When you transform a random variable using a formula like Y = a + bX, the new mean becomes a + b? and the new standard deviation becomes |b|?, illustrating how deterministic adjustments affect the overall distribution.

Expected Value

The expected value, or mean, of a random variable is the long-run average value it takes. In the case of linear transformations, the expected value of the transformed variable is computed by applying the transformation to the original mean, which allows for easy calculation of shifts resulting from constant additions and proportional changes.

Standard Deviation

Standard deviation measures the spread of a random variable's values around the mean. When a random variable undergoes a linear transformation, its standard deviation is multiplied by the absolute value of the scaling constant, indicating how changes in scale directly affect the variability of the outcomes.

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