34,707 Pages

In the study of probability, given two random variables X and Y, the joint distribution of X and Y is the distribution of X and Y together.

## The discrete case

For discrete random variables, the joint probability mass function can be written as Pr(X = x & Y = y). This is

Since these are probabilities, we have

## The continuous case

Similarly for continuous random variables, the joint probability density function can be written as fX,Y(xy) and this is

where fY|X(y|x) and fX|Y(x|y) give the conditional distributions of Y given X = x and of X given Y = y respectively, and fX(x) and fY(y) give the marginal distributions for X and Y respectively.

Again, since these are probability distributions, one has

## Joint distribution of independent variables

If for discrete random variables for all x and y, or for continuous random variables for all x and y, then X and Y are said to be independent.

## Multidimensional distributions

The joint distribution of two random variables can be extended to many random variables X1, ..., Xn by adding them sequentially with the identity