Mean and variance of linear function of a random variable

Let X be a random variable that takes the value x with probability P (x) and consider a new random variable Y, defined by Y = a + bX.

Suppose that random variable X has mean , and variance .

Then mean and variance of Y are

and

so that standard deviation of Y is

.

Example:

A car salesman estimates the following probabilities for the number of cars that he will sell in next month.

Number cars	0 1 2 3 4
Probability	0.12 0.20 0.25 0.25 0.18

a) Find the expected number of cars that will be sold in the next month.

b) Find the standard deviation of the number of cars that will be sold in

next month.

c) The salesperson receives for the month a salary of $300, plus an additional $200 for each car sold. Find the mean and standard deviation of his total monthly salary.

Solution:

a) The random variable X has mean

.

b)Variance 1.621

.

c) Total monthly salary of salesperson can be written as . Then

$734.

.

=$254.64.

Summary results for the mean and variance of special linear functions:

a) Let b =0 in the linear function, . Then for any constant a.

and Var (a) =0

If a random variable always takes the value a, it will have a mean a and

a variance 0.

b) Let in the linear function, . Then .

and .

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