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The theoretical definition for marginal distribution can easily be drawn, but

Why/where are marginal distributions used?

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    $\begingroup$ Any time you are working with more than one random variable and you consider the distribution of just a single one (or, more generally, the joint distribution of a proper subset of them) then that is a marginal distribution. In light of that, I think this question answers itself. $\endgroup$ – whuber Nov 9 '15 at 14:13
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For example: if you are a planner and are deciding how many school classes there should be in a particular region you care about the expected number of students, not the expected number of students conditional on family background, gender, ... So that would be a case were the marginal distribution is relevant (though, strictly speaking, it is still conditional on region).

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