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Open AccessArticle

Some Useful Integral Representations for Information-Theoretic Analyses

The Andrew and Erna Viterbi Faculty of Electrical Engineering, Israel Institute of Technology Technion City, Haifa 3200003, Israel
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Entropy 2020, 22(6), 707; https://doi.org/10.3390/e22060707
Received: 13 May 2020 / Revised: 9 June 2020 / Accepted: 24 June 2020 / Published: 26 June 2020
(This article belongs to the Special Issue Information Theory for Communication Systems)
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 nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive non-integer real (also known as fractional moments). 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 nonnegative 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. View Full-Text
Keywords: integral representation; logarithmic expectation; moment-generating function; fractional moments; Rényi entropy; jamming; estimation errors; multivariate Cauchy distributions; guessing integral representation; logarithmic expectation; moment-generating function; fractional moments; Rényi entropy; jamming; estimation errors; multivariate Cauchy distributions; guessing
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Merhav, N.; Sason, I. Some Useful Integral Representations for Information-Theoretic Analyses. Entropy 2020, 22, 707.

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