论文标题
信息理论分析的一些有用的积分表示
Some Useful Integral Representations for Information-Theoretic Analyses
论文作者
论文摘要
这项工作是我们先前文章的扩展，其中探索了对数函数的众所周知的积分表示，并伴随着其在获得紧凑，易于估计的，准确的公式的证明，以涉及涉及正随机变量对数的期望。在这里，我们以同样的精神得出了非负随机变量的矩或这种独立随机变量的总和的确切积分表示（在一个或两个维度上），其中力矩顺序是一般的非局体真实（也称为分数时刻）。所提出的公式应用于具有信息理论动机的各种示例，并显示其如何促进其数值评估。特别是，当应用于非负随机变量的大数字总和的计算时，很明显，正如我们所提出的积分表示所建议的那样，比在$ n $尺寸上集成的选择要容易得多，这是要在$ n $ dimensions of $ n $ dimensions上的替代方案，这是所需的n $ dimensions的替代方案。
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.
