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 noninteger 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.Comment: Published in Entropy, vol. 22, no. 6, paper 707, pages 1-29, June
2020. Available at: https://www.mdpi.com/1099-4300/22/6/70
