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Joint distribution
 
 
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In the study of probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
, given two random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
s X and Y, the joint distribution of X and Y defines the probability of events defined in terms of both X and Y. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number random variables, giving a multivariate distribution.








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In the study of probability
Probability

Probability, or wikt:chance, is a way of expressing knowledge or belief that an Event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about t...
, given two random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
s X and Y, the joint distribution of X and Y defines the probability of events defined in terms of both X and Y. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number random variables, giving a multivariate distribution.

The general case


The joint probability distribution of a pair of random variables is defined by the joint cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
;

Similarly, the joint distribution of a multivariate distribution is defined by the joint cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 for the set of random variables.

The discrete case


For discrete random variables, the joint probability mass function
Probability mass function

In probability theory, a probability mass function is a function that gives the probability that a discrete random variable random variable is exactly equal to some value....
 is

Since these are probabilities, we have

The continuous case


Similarly for continuous random variables, the joint probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 can be written as fX,Y(x, y) and this is

where fY|X(y|x) and fX|Y(x|y) give the conditional distribution
Conditional distribution

Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value....
s of Y given X = x and of X given Y = y respectively, and fX(x) and fY(y) give the marginal distribution
Marginal distribution

In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset....
s for X and Y respectively.

Again, since these are probability distributions, one has

Joint distribution of independent variables


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
Statistical independence

In probability theory, to say that two event s are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs....
.

Multidimensional distributions

The joint distribution of two random variables can be extended to many random variables X1, ..., Xn by adding them sequentially with the identity

See also


  • Chow-Liu tree
    Chow-Liu tree

    A Chow-Liu tree is an efficient method for constructing a second-order product approximation of a joint distribution, first described in a paper by ....
  • Copula (statistics)
    Copula (statistics)

    In statistics, a copula is used as a general way of formulating a Joint probability distribution#Multidimensional distributions in such a way that various general types of dependence can be represented....
  • Bayesian networks
  • Statistical interference
    Statistical interference

    When two probability distributions overlap, statistical interference exists. Knowledge of the distributions can be used to determine the likelihood that one parameter exceeds another, and by how much....


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