Mean and Variance of Random Variables
Themeanof a discrete random variableXis 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 theexpected valueofX)
is , formally defined by
The mean of a random variable provides thelong-run averageof the variable, or the expected average outcome over many observations.
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.
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