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The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables and the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s).
Joint and marginal distributions of a pair of discrete random variables, X and Y, dependent, thus having nonzero mutual information I(X; Y). The values of the joint distribution are in the 3×4 rectangle; the values of the marginal distributions are along the right and bottom margins.
where the marginal, joint, and/or conditional probability density functions are denoted by with the appropriate subscript. This can be simplified as
The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by the Boltzmann constant).
The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable. If the conditional distribution of Y {\displaystyle Y} given X {\displaystyle X} is a continuous distribution , then its probability density function is known as the ...
To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.
Your marginal tax rate is the rate of tax you pay on the portion of your income that falls in the highest tax bracket that applies to you. The IRS adjusts its tax brackets for inflation annually.
Given a model of the joint distribution, (,), the distribution of the individual variables can be computed as the marginal distributions = (, =) and () = (, =) (considering X as continuous, hence integrating over it, and Y as discrete, hence summing over it), and either conditional distribution can be computed from the definition of conditional ...