The expected value or mean of a probability function of a random variable gives a measure of the average or central tendency, but no indication of how spread the values are likely to be.
The standard deviation or variance of a probability distribution is measure of the spread of likely values of a random variable.
The variance of the discrete random variable X with E(X) = μ is given by:
|
VAR(X)
|
= | E(X − μ)2 |
| = | Σ(x − μ)2 .P(x) |
Example
The probability distribution table for the random variable X is shown below.
|
x
|
3
|
4
|
5
|
6
|
7
|
|
P(X = x)
|
0.1
|
0.2
|
0.4
|
0.2
|
0.1
|
The expected value E(X) = 5 (because the distribution is symmetrical)
The variance VAR(X) = Σ(x − μ)2 .P(x)
|
x
|
3
|
4
|
5
|
6
|
7
|
|
x − 5
|
3 − 5 = -2
|
4 − 5 = -1
|
5 − 5 = 0
|
6 − 5 = 1
|
7 − 5 = 2
|
|
(x − 5)2
|
(-2)2 = 4
|
(-1)2 = 1
|
02 = 0
|
12 = 1
|
22 = 4
|
|
P(X = x)
|
0.1
|
0.2
|
0.4
|
0.2
|
0.1
|
VAR(X) = 4 x 0.1 + 1 x 0.2 + 0 x 0.4 + 1 x 0.2 + 4 x 0.1
= 1.2
VAR(X) = 1.2
Alternative Formula for the Variance of a Random Variable
The formula used above requires quite a lot of calculation for each value of the variable.
The following formula is quicker to use:
|
VAR(X) = E(X2) – [E(X)]2
|
Proof
|
VAR(X)
|
= E(X − μ)2 |
| = E(X2 − 2μX + μ2) | |
| = E(X2) − 2μE(X) +E(μ2) | |
|
= E(X2) − 2μ2 + μ2 |
|
| = E(X2) − μ2 | |
| = E(X2) − [E(X)]2 |
Example
The probability distribution table for the random variable X is shown below.
|
x
|
3
|
4
|
5
|
6
|
7
|
|
P(X = x)
|
0.1
|
0.2
|
0.4
|
0.2
|
0.1
|
The expected value E(X) = 5 (because the distribution is symmetrical)
E(X2) = 9 x 0.1 + 16 x 0.2 + 25 x 0.4 + 36 x 0.2 + 49 x 0.1
= 26.2
VAR(X) = E(X2) − [E(X)]2
= 26.2 − 52
= 1.2
VAR(X) = 1.2
Summary
The following notation is usually used in statistics and probability:
|
Mean
|
Standard deviation
|
Variance
|
|
|
Random variables |
E(X)
|
SD(X)
|
VAR(X)
|
|
Sample |
 |
s
|
s2
|
|
Population |
μ
|
σ
|
σ2
|
