Joint distribution.html

In the study of probability, given two random variables X and Y, the joint distribution of X and Y is the distribution of the intersection of the events X and Y, that is, of both events X and Y occurring together. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of events or random variables, giving a multivariate distribution. The joint probability of X and Y is written $\scriptstyle P(X \cap Y)$ or $\scriptstyle P(X, Y).$

1 The discrete case
2 The continuous case
3 Joint distribution of independent variables
4 Multidimensional distributions
5 See also
6 External links

The discrete case

For discrete random variables, the joint probability mass function is

$\begin{align} \mathrm{P}(X=x\ \mathrm{and}\ Y=y) & {} = \mathrm{P}(Y=y \mid X=x) \cdot \mathrm{P}(X=x) \\ & {} = \mathrm{P}(X=x \mid Y=y) \cdot \mathrm{P}(Y=y). \end{align}$

Since these are probabilities, we have

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

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.

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, ..., X_n by adding them sequentially with the identity

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

External links

Joint continuous density function on PlanetMath