Joint distribution

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


 * $$P(X=x\ \mathrm{and}\ Y=y) = P(Y=y|X=x)P(X=x)= P(X=x|Y=y)P(Y=y).\;$$

Since these are probabilities, we have


 * $$\sum_x \sum_y P(X=x\ \mathrm{and}\ Y=y) = 1.\;$$

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


 * $$f_{X,Y}(x,y) = f_{Y|X}(y|x)f_X(x) = f_{X|Y}(x|y)f_Y(y)$$

where fY(y|x) and fX(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
 * $$\int_x \int_y f_{X,Y}(x,y) \; dy \; dx= 1.$$

Joint distribution of independent variables
If for discrete random variables $$\ P(X = x \ \mbox{and} \ Y = y ) = P( X = x) \cdot P( Y = y) $$ for all x and y, or for continuous random variables $$\ p_{X,Y}(x,y) = p_X(x) \cdot p_Y(y) $$ 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


 * $$f_{X_1, \ldots, X_n}(x_1, \ldots, x_n) = f_{X_n | X_1, \ldots, X_{n-1}}( x_n | x_1, \ldots, x_{n-1}) f_{X_1, \ldots, X_{n-1}}( x_1, \ldots, x_{n-1} ) .$$