**Mean and Variance of Random Variables**

**Mean**

The* mean*of a discrete random variable

*X*is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome

*xi*according to its probability,

*pi*. The common symbol for the mean (also known as the

*of*

**expected value***X*)

is , formally defined by

The mean of a random variable provides the*long-run average*of the variable, or the expected
average outcome over many observations.

**Example**

Suppose an individual plays a gambling game where it is possible to lose $1.00, break even, win $3.00, or win $10.00 each time she plays. The probability distribution for each outcome is provided by the following table:

Outcome -$1.00 $0.00 $3.00 $5.00 Probability 0.30 0.40 0.20 0.10

The mean outcome for this game is calculated as follows:

= (-1*.3) + (0*.4) + (3*.2) + (10*0.1) = -0.3 + 0.6 + 0.5 = 0.8.

In the long run, then, the player can expect to win about 80 cents playing this game -- the odds are in her favor.

**Probability Distributions**

A listing of all the values the random variable can assume with their corresponding probabilities make a probability distribution.

A note about random variables. A random variable does not mean that the values can be anything (a random number). Random variables have a well defined set of outcomes and well defined probabilities for the occurrence of each outcome. The random refers to the fact that the outcomes happen by chance -- that is, you don't know which outcome will occur next.

Here's an example probability distribution that results from the rolling of a single fair die.

**x** **1** **2** **3** **4** **5** **6** **sum**