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

<math>P(X=x\ \mathrm{and}\ 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\ \mathrm{and}\ Y=y) = 1.\;</math>

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

<math>f_{X,Y}(x,y) = f_{Y|X}(y|x)f_X(x) = f_{X|Y}(x|y)f_Y(y)</math>

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.

Again, since these are probability distributions, one has

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

Joint distribution of independent variables

If for discrete random variables <math>\ P(X = x \ \mbox{and} \ Y = y ) = P( X = x) \cdot P( Y = y) </math> for all x and y, or for continuous random variables <math>\ p_{X,Y}(x,y) = p_X(x) \cdot p_Y(y) </math> 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, ..., X_n by adding them sequentially with the identity

<math>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} ) .</math>

See also

• Chow-Liu Tree
• Copula (statistics)
• Bayesian networks

External links

• Joint continuous density function on PlanetMathko:결합 분포