Independence
If the conditional probabilities for Y are the same for all values of X, then Y is said to be independent of X.
If X and Y are independent, knowing the value of X does not give us any information about the likely value for Y.
Example
An example of independence is given by the following table of joint probabilities for the weight category and work performance (as assessed by a supervisor) of supermarket employees.
Work performance | ||||
---|---|---|---|---|
Poor | Satisfactory | Above average | Marginal | |
Underweight | 0.0225 | 0.1125 | 0.0150 | 0.1500 |
Normal | 0.0825 | 0.4125 | 0.0550 | 0.5500 |
Overweight | 0.0300 | 0.1500 | 0.0200 | 0.2000 |
Obese | 0.0150 | 0.0750 | 0.0100 | 0.1000 |
Marginal | 0.1500 | 0.7500 | 0.1000 | 1.0000 |
The proportional Venn diagram for this model is shown below.
The conditional probability of above average work performance is the same for all weight categories — knowing an employee's weight would not help you to predict their work performance. The two variables are therefore independent.
Mathematical definition of independence
If Y is independent of X, then: