A salesperson receives an annual salary of $6,000 plus 8% of the value of the orders she takes. The annual value of these orders can be represented by a random variable with a mean of $600,000 and a standard deviation of $180,000. Find the mean and standard deviation of the salesperson’s annual income.

Explanation:

1. Mean of the annual income:

The mean income is the expected income, which is: the sum of the annual salary (constant) plus the 8% of the mean value of the orders ($600,000):

2. Standard deviation of the annual income.

The standar deviation is a measure of  how extended the values are.

It means that the annual value of the orders will be around the mean plus or minus a number of standard deviations, depending on the precision you want.

The 8% of the the standard deviation is 8% ×  $180,000 = $14,400.

Since the $6,000 is a constant it does not modify the standard deviation.

These results are a consequence of the linearity of the mean and the standard deviation.

Call Y the salesperson salary, and X the valueof the orders. Then:

The linearity property states that:

And:

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