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.

Independence implies that the sub-populations corresponding to different values of X all contain values of Y in the same proportions.

Mathematical performance and weight

As an example of independence, we continue with the (artificial) example on the previous page. We now show the relationship between weight and performance in a mathematics test. In this model, weight and performance are independent — knowing someone's weight gives no clues as to that person's mathematical ability.


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

For this model, the conditional probabilities for mathematical performance, given weight, are:

Conditional Probabilities
Mathematical performance
    Poor     Satisfactory Above average Total
Underweight 0.15 0.75 0.10 1.0
Normal 0.15 0.75 0.10 1.0
Overweight 0.15 0.75 0.10 1.0
Obese 0.15 0.75 0.10 1.0

The conditional probabilities are the same for each weight, so knowing that a student is, say, obese does not affect the probability of above-average in mathematics. The proportional Venn diagram has the form shown below.

Note that the Proportional Venn Diagram now consists of a grid of horizontal and vertical lines.

Mathematical definition of independence

If Y is independent of X, then:

Also, if Y is independent of X, then X is also independent of Y.

Since the conditional and marginal probabilities are equal if Y and X are independent, an equivalent definition of independence is:

X and Y are independent if      pxy  =  px × py