  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.
