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Marginal probability density function [ edit ] Given two continuous random variables X and Y whose joint distribution is known, then the marginal probability density function can be obtained by integrating the joint probability distribution, f , over Y, and vice versa.
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...
where (,) is the copula density function, () and () are the marginal probability density functions of X and Y, respectively. There are four elements in this equation, and if any three elements are known, the fourth element can be calculated.
Formally, , (,) is the probability density function of (,) with respect to the product measure on the respective supports of and . Either of these two decompositions can then be used to recover the joint cumulative distribution function:
The logarithm must be taken to base e since the two terms following the logarithm are themselves base-e logarithms of expressions that are either factors of the density function or otherwise arise naturally. The equation therefore gives a result measured in nats. Dividing the entire expression above by log e 2 yields the divergence in bits.
where (,) is the copula density function, () and () are the marginal probability density functions of X and Y, respectively. It is important understand that there are four elements in the equation 1, and if three of the four are know, the fourth element can me calculated.
This can be derived using the exponential family formula for the moment generating function of the ... , then the marginal ... density function ...
If the conditional distribution of given is a continuous distribution, then its probability density function is known as the conditional density function. [1] The properties of a conditional distribution, such as the moments , are often referred to by corresponding names such as the conditional mean and conditional variance .