Mean and Variance of Random Variables
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 expected value of 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
Probability
0.30
-$1.00 $0.00
0.40
0.20
$3.00
0.10
$5.00
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
p(x)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
sum
6/6=1
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 expected value of 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
Probability
0.30
-$1.00 $0.00
0.40
0.20
$3.00
0.10
$5.00
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
p(x)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
sum
6/6=1