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There are many problems where one must reduce the dimension of a large system of equations before one can even start solving it on the computer. Dimensional reduction is also an aspect of renormalization group analysis and the computation of critical exponents in physics. A useful technique of dimensional reduction and renormalization are connected by a theorem due to Hald which asserts that dimensional reduction by conditional expectation leaves invariant the marginal distributions of the variables that are kept. The theorem has an elementary proof; and I will present everything from scratch.