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Wikipedia / Joint probability distribution

Given random variables, that are defined on the same probability space, the multivariate or joint probability distribution is a probability distribution 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 variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the 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.

Examples

Draws from an urn

be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively. The probability of drawing a red ball from either of the urns is , and the probability of drawing a blue ball is . The joint probability distribution is presented in the following table:Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are the joint distribution. In any one cell the probability of a particular combination occurring is (since the draws are independent) the product of the probability of the specified result for A and the probability of the specified result for B. The probabilities in these four cells sum to 1, as with all probability distributions.

Moreover, the final row and the final column give the marginal probability distribution for A and the marginal probability distribution for B respectively. For example, for A the first of these cells gives the sum of the probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, as 's probabilities unconditional, in a margin of the table.

Coin flips

Consider the flip of two fair coins be discrete random variables associated with the outcomes of the first and second coin flips respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution. If a coin displays "heads" then the associated random variable takes the value 1, and it takes the value 0 otherwise. The probability of each of these outcomes is , so the marginal (unconditional) density functions are

Rolling a die

Consider the roll of a fair die otherwise.

These probabilities necessarily sum to 1, since the probability of some occurring is 1.

Marginal probability distribution

If more than one random variable is defined in a random experiment, it is important to distinguish between the joint probability distribution of X and Y and the probability distribution of each variable individually. The individual probability distribution of a random variable is referred to as its marginal probability distribution. In general, the marginal probability distribution of X can be determined from the joint probability distribution of X and other random variables.

, the marginal probability density function of X and Y, which defines the marginal distribution, is given by:

where the first integral is over all points in the range of (X,Y) for which X=x and the second integral is over all points in the range of (X,Y) for which Y=y.

Joint cumulative distribution function

is given by

where the right-hand side represents the probability and.

is given by

random variables as a random vector yields a shorter notation:

Joint density function or mass function

Discrete case

The joint probability mass function of two discrete random variables is:

is the probability.

which is:

or equivalently

This identity is known as the chain rule of probability.

Since these are probabilities, in the two-variable case

to

Continuous case

The joint probability density function for two continuous random variables is defined as the derivative of the joint cumulative distribution function (see ):

are the conditional distributions are the marginal distributions respectively.

The definition extends naturally to more than two random variables:

Mixed case

One example of a situation in which one may wish to find the cumulative distribution of one random variable which is continuous and another random variable which is discrete arises when one wishes to use a logistic regression. One must with respect to the product measure on the respective supports The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables.

Additional properties

Joint distribution for independent variables

are independent

.

While the number of independent random events grows, the related joint probability value decreases rapidly to zero, according to a negative exponential law.

. This means that acquiring any information about the value of one or more of the random variables leads to a conditional distribution of any other variable that is identical to its unconditional (marginal) distribution; thus no variable provides any information about any other variable.

Joint distribution for conditionally dependent variables

is conditionally dependent. Such conditional independence relations can be represented with a Bayesian network or copula functions.

Covariance

When two or more random variables are defined on a probability space, it is useful to describe how they vary together; that is, it is useful to measure the relationship between the variables. A common measure of the relationship between two random variables is the covariance. Covariance is a measure of linear relationship between the random variables. If the relationship between the random variables is nonlinear, the covariance might not be sensitive to the relationship, which means, it does not relate the correlation between two variables.

is

Correlation

There is another measure of the relationship between two random variables that is often easier to interpret than the covariance.

The correlation just scales the covariance by the product of the standard deviation of each variable. Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρXY is near +1 (or −1). If ρXY equals +1 or −1, it can be shown that the points in the joint probability distribution that receive positive probability fall exactly along a straight line. Two random variables with nonzero correlation are said to be correlated. Similar to covariance, the correlation is a measure of the linear relationship between random variables.

Important named distributions

Named joint distributions that arise frequently in statistics include the multivariate normal distribution, the multivariate stable distribution, the multinomial distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution.

See also

References