Given two random variables X and Y, the joint probability distribution of X and Y is the probability distribution of X and Y together.

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

<math>P(X=x,Y=y) = P(Y=y|X=x)P(X=x)= P(X=x|Y=y)P(Y=y).\;<math>

Since these are probabilities, we have

<math>\sum_x \sum_y P(X=x,Y=y) = 1.\;<math>

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

<math>p_{X,Y}(x,y)=p_{Y|X}(y|x)p_X(x) = p_{X|Y}(x|y)p_Y(y) \;<math>

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

Since this is a probability density, we have

<math>\int_x \int_y p_{X,Y}(x,y) \; dy \; dx= 1.<math>

If for discrete random variables P(X=x,Y=y)=P(X=x)P(Y=y) for all x and y, or for continuous random variables p_X,Y(x,y)=p_X(x)p_Y(y) for all x and y, then X and Y are said to be independent.