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What are the relationships between correlation and conditional distribution. For instance, given three dependent variables, X1, X2 and X3. If their respective correlation matrix is known, are there ways to get the conditional distribution, such as P(X1|X2) or P(X1|X2,X3)?

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No. The correlation is a summary statistic, while (conditional, joint) distributions involve the complete information about how the three random variables are related. But, you can generate the correlation matrix from the joint distribution. Your query is only possible in very speciaş cases, e.g. when variables are jointly normal, with known means and variances.

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