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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. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables.
In a model where a Dirichlet prior distribution is placed over a set of categorical-valued observations, the marginal joint distribution of the observations (i.e. the joint distribution of the observations, with the prior parameter marginalized out) is a Dirichlet-multinomial distribution.
In general, the marginal probability distribution of X can be determined from the joint probability distribution of X and other random variables. If the joint probability density function of random variable X and Y is , (,), the marginal probability density function of X and Y, which defines the marginal distribution, is given by: =, (,)
The mutual information of two multivariate normal distribution is a special case of the Kullback–Leibler divergence in which is the full dimensional multivariate distribution and is the product of the and dimensional marginal distributions and , such that + =.
Marginal or peripheral populations are those found at the boundary of the range. [5] [6] [7] When the distribution of a species is changing, the leading edge populations are at the expanding geographic edge of the distribution range whilst rear edge populations are undergoing retreat. [5]
In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe/model the dependence (inter-correlation) between random variables. [1]
The Dirichlet distribution is a conjugate distribution to the negative multinomial distribution. This fact leads to an analytically tractable compound distribution.For a random vector of category counts = (, …,), distributed according to a negative multinomial distribution, the compound distribution is obtained by integrating on the distribution for p which can be thought of as a random ...
The distributions of each of the component random variables are called marginal distributions. The conditional probability distribution of given is the probability distribution of when is known to be a particular value.