joint distribution

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 f_X,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 f_Y|X(y|x) and f_X|Y(x|y) give the conditional distributions of Y given X = x and of X given Y = y respectively, and f_X(x) and f_Y(y) give the marginal distributions for X and Y respectively. Since this is a probability density, we have

\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. The joint distribution of two random variables can be extended to many random variables X₁, ..., X_n 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} ) .

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