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 employee's total pay in a week.

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Key Concepts

Expected Value

The expected value (or mean) of a random variable is the long-run average value it takes over numerous samples. It is calculated as a weighted average of all possible values, and it represents the central or typical outcome of the random process.

Standard Deviation

The standard deviation measures the dispersion or variability of the random variable's values around its mean. It is the square root of the variance and provides a quantitative measure of how spread out the outcomes are from their average.

Linear Transformation of Random Variables

A linear transformation of a random variable involves making adjustments by scaling and/or shifting its values. When a new variable is defined as a constant plus a multiple of the original variable (Y = a + bX), the expected value transforms as E[Y] = a + bE[X] and the standard deviation scales as SD(Y) = |b|SD(X). This principle is essential for understanding how changes to the random variable affect its distribution.

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