Random quantities often occur together. A joint distribution describes the uncertainty of two or more random variables at the same time.

For a pair $(X,Y)$, probabilities can concern both variables, such as

$$P(X=x,Y=y)$$

or events such as

$$P(X\le a,,Y>b).$$

Joint probability tables

For discrete variables, a table can assign a probability to every pair of possible values. Each cell contains

$$P(X=x,Y=y),$$

and all cells together sum to $1$.

Marginal distributions

The distribution of one variable alone is recovered by summing over the possible values of the other. For example,

$$P(X=x)=\sum_y P(X=x,Y=y).$$

These one-variable distributions are called marginal distributions.

Conditional distributions

Once $Y=y$ is known, the distribution of $X$ may change. For $P(Y=y)>0$,

$$P(X=x\mid Y=y) =\frac{P(X=x,Y=y)}{P(Y=y)}.$$

This is ordinary conditional probability applied to random-variable values.

Independence of random variables

Two discrete random variables are independent when their joint probabilities factor:

$$P(X=x,Y=y)=P(X=x)P(Y=y)$$

for every pair of values.

A joint distribution therefore contains information that the two marginal distributions alone cannot reveal: it describes how the variables vary together.