Some Useful Integral Representations for Information-Theoretic Analyses
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This work is an extension of our earlier article, where a well–known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily–calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a non–negative random variable, or the sum of such independent random variables, where the moment order is a general positive real, not necessarily an integer. The proposed formula is applied to a variety of examples with an information–theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, n, of non–negative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over n dimensions, as needed in the direct calculation of the desired moment.
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