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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 caseEdit

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 caseEdit

Similarly for continuous random variables, the **joint probability density function** can be written as *f*_{X,Y}(*x*, *y*) and this is

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 Edit

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 distributionsEdit

The joint distribution of two random variables can be extended to many random variables *X*_{1}, ..., *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} ) . $

## See alsoEdit

## External linksEdit

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