# Joint distribution

## Redirected from Joint probability distribution

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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

Since these are probabilities, we have

## 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

## Joint distribution of independent variables Edit

If for discrete random variables for all *x* and *y*, or for continuous random variables 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

## See alsoEdit

## External linksEdit

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