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The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out.
Thus, a more detailed study would examine changes to the whole of the marginal distribution. An intermediate-level study might move from looking at the variability to studying changes in the skewness. In addition to these, questions of homogeneity apply also to the joint distributions.
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: =, (,)
A frequency distribution shows a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data notably to show results of an election, income of people for a certain region, sales of a product within a certain period, student loan amounts of graduates, etc.
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.
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 + =.
Mutual information is a measure of the inherent dependence expressed in the joint distribution of and relative to the marginal distribution of and under the assumption of independence. Mutual information therefore measures dependence in the following sense: I ( X ; Y ) = 0 {\displaystyle \operatorname {I} (X;Y)=0} if and only if X ...
An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for ...