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 mathematical ability of high school children.


Joint Probabilities
Mathematical 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 maths performance is the same for all weight categories — knowing a child's weight would not help you to predict maths performance. The two variables are therefore independent.

Mathematical definition of independence

If Y is independent of X, then: