Editing Joint probability distribution (section)

{{Short description|Type of probability distribution}}
{{Use dmy dates|date=January 2021}}
{{Annotated image | caption=Many sample observations (black) are shown from a joint probability distribution. The marginal densities are shown as well (in blue and in red).
| image=Multivariate normal sample.svg
| image-width = 300
| annotations =
{{Annotation|60|190|<math>X</math>}}
{{Annotation|240|190|<math>Y</math>}}
{{Annotation|240|20|<math>p(X)</math>}}
{{Annotation|60|20|<math>p(Y)</math>}}
}}

{{Probability fundamentals}}
Given [[random variable]]s <math>X,Y,\ldots</math>, that are defined on the same<ref>{{Cite book | author = Feller, William  | title = An introduction to probability theory and its applications, vol 1, 3rd edition | date=1957 | pages = 217-218 | ISBN = 978-0471257080 | language = en }}</ref> [[probability space]], the '''multivariate''' or '''joint probability distribution''' for <math>X,Y,\ldots</math> is a [[probability distribution]] that gives the probability that each of <math>X,Y,\ldots</math> falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a [[bivariate distribution]], but the concept generalizes to any number of random variables.

The joint probability distribution can be expressed in terms of a joint [[cumulative distribution function]] and either in terms of a joint [[probability density function]] (in the case of [[continuous variable]]s) or joint [[probability mass function]] (in the case of [[Discrete probability distribution|discrete]] variables). These in turn can be used to find two other types of distributions: the [[marginal density|marginal distribution]] giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the [[conditional probability distribution]] giving the probabilities for any subset of the variables conditional on particular values of the remaining variables.