<P> Marginal variables are those variables in the subset of variables being retained . These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table . 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 . </P> <P> The context here is that the theoretical studies being undertaken, or the data analysis being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables . In many applications, an analysis may start with a given collection of random variables, then first extend the set by defining new ones (such as the sum of the original random variables) and finally reduce the number by placing interest in the marginal distribution of a subset (such as the sum). Several different analyses may be done, each treating a different subset of variables as the marginal variables . </P> <P> Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is simply the probability distribution of X averaging over information about Y . It is the probability distribution of X when the value of Y is not known . This is typically calculated by summing or integrating the joint probability distribution over Y . </P> <Table> <Tr> <Th> X Y </Th> <Th> x </Th> <Th> x </Th> <Th> x </Th> <Th> x </Th> <Th> p (Y) ↓ </Th> </Tr> <Tr> <Th> y </Th> <Td> ⁄ </Td> <Td> ⁄ </Td> <Td> ⁄ </Td> <Td> ⁄ </Td> <Th> ⁄ </Th> </Tr> <Tr> <Th> y </Th> <Td> ⁄ </Td> <Td> ⁄ </Td> <Td> ⁄ </Td> <Td> ⁄ </Td> <Th> ⁄ </Th> </Tr> <Tr> <Th> y </Th> <Td> ⁄ </Td> <Td> ⁄ </Td> <Td> ⁄ </Td> <Td> ⁄ </Td> <Th> ⁄ </Th> </Tr> <Tr> <Th> y </Th> <Td> ⁄ </Td> <Td> 0 </Td> <Td> 0 </Td> <Td> 0 </Td> <Th> ⁄ </Th> </Tr> <Tr> <Th> p (X) → </Th> <Th> ⁄ </Th> <Th> ⁄ </Th> <Th> ⁄ </Th> <Th> ⁄ </Th> <Th> ⁄ </Th> </Tr> <Tr> <Td_colspan="6"> Joint and marginal distributions of a pair of discrete, random variables X, Y having nonzero mutual information I (X; Y). The values of the joint distribution are in the 4 × 4 square, and the values of the marginal distributions are along the right and bottom margins . </Td> </Tr> </Table>

How to calculate marginal distribution from joint distribution
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