Because one primary objective of econometrics is to examine relationships between variables, you need to be familiar with probabilities that combine information on two variables. A bivariate or joint probability density provides the relative frequencies (or chances) that events with more than one random variable will occur. Generally, this information is shown in a table.
For two random variables, X and Y, you’re already familiar with the notation for joint probabilities from your statistics class, which uses the intersection term like this:
The variables a and b are possible values for the random variable. However, in econometrics, you likely need to become familiar with this mathematical notation for joint probabilities: f(X, Y). In this notation, the comma is used instead of the intersection operator.
The table provides an example of a joint probability table for random variables X and Y. The column headings in the middle of the first row list the X values (1, 2, and 3), and the first column lists the Y values (1, 2, 3, and 4). The values contained in the middle represent the joint or intersection probabilities.
For example, the probability X equals 3 (see column 3) and Y equals 2 (row 2) is 0.10. In your econometrics class, the mathematical notation used to express this is likely to look like f(X = 3, Y = 2) = 0.10.
| Y | X | f (Y) | ||
|---|---|---|---|---|
| 1 | 2 | 3 | ||
| 1 | 0.25 | 0 | 0.10 | 0.35 |
| 2 | 0.05 | 0.05 | 0.10 | 0.20 |
| 3 | 0 | 0.05 | 0.20 | 0.25 |
| 4 | 0 | 0 | 0.20 | 0.20 |
| f(X) | 0.30 | 0.10 | 0.60 | 1.00 |
You can also see that the column sums, f(X), contain the marginal or unconditional probabilities for random variable X and the row sums, f(Y), contain the same information for random variable Y. For example, f(Y = 3) = 0.25; that is, the probability that Y equals 3 is 0.25.
