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It is well-known that maximum entropy distributions, subject to appropriate moment constraints, arise in physics and mathematics. In an attempt to find a physical reason for the appearance of maximum entropy distributions, the following theorem is offered. The conditional distribution of given the empirical observation , where are independent identically distributed random variables with common density converges to (Suitably normalized), where is chosen to satisfy . Thus the conditional distribution of a given random variable is the (normalized) product of the maximum entropy distribution and the initial distribution. This distribution is the maximum entropy distribution when is uniform. The proof of this and related results relies heavily on the work of Zabell and Lanford.