Bivariate or Joint Probability Density and Econometrics

By Roberto Pedace

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.

Joint Probability Table
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.