Moments and Absolute Moments of the Normal Distribution (1209.4340v2)

Abstract: We present formulas for the (raw and central) moments and absolute moments of the normal distribution. We note that these results are not new, yet many textbooks miss out on at least some of them. Hence, we believe that it is worthwhile to collect these formulas and their derivations in these notes.

• The paper's main contribution is the comprehensive derivation of raw, central, and absolute moments for the normal distribution, consolidating dispersed literature.
• It employs special functions, including the gamma and parabolic cylinder functions, to derive precise moment formulas.
• The findings enhance statistical modeling and inference, with applications in simulation studies, econometrics, and machine learning.

Insights into Moments of the Normal Distribution

The paper, "Moments and Absolute Moments of the Normal Distribution," authored by Andreas Winkelbauer, provides a comprehensive examination of various moment-related formulas for the normal distribution, also known as the Gaussian distribution. Despite the prevalence of these formulas in existing statistical literature, the paper argues that they are often scattered across multiple sources or omitted altogether. Thus, it presents a consolidated derivation of these formulas to facilitate ease of reference for statistical researchers.

The normal distribution, denoted as $X \sim \mathcal{N}(\mu, \sigma^2)$, is a fundamental construct in statistics, characterized by its bell curve shape determined by its mean $\mu$ and variance $\sigma^2$. The paper focuses on four types of moments: raw moments, central moments, raw absolute moments, and central absolute moments, all calculated as expectations over powers or absolute powers of the variable $X$. These moments are crucial in statistical analysis as they provide insights into properties such as shape, variability, and expected behavior of the distribution in question.

Numerical Results

1. Raw Moments: The expectation $\mathrm{E}\{X^\nu\}$ is derived in terms of parabolic cylinder functions, notably involving the imaginary unit, the gamma function, and the confluent hypergeometric functions. This demonstrates the intricate nature of raw moments beyond basic quadratic calculations often seen in applied settings.
2. Central Moments: The paper reaffirms that only even central moments are non-zero for a Gaussian distribution, confirming the inherent symmetry of the distribution about its mean. The expectation $\mathrm{E}\{(X-\mu)^\nu\}$ reduces neatly to zero for odd $\nu$.
3. Absolute Moments: Raw and central absolute moments, represented as $\mathrm{E}\{\lvert X \rvert^\nu\}$ and $\mathrm{E}\{\lvert X - \mu \rvert^\nu\}$ respectively, are particularly useful in fields like robust statistics where absolute deviations often substitute squared deviations to mitigate the influence of outliers.

The paper presents meticulous derivations of these formulas, guided by integrals and special functions such as the gamma function, Kummer's functions, and parabolic cylinder functions. This not only underscores the mathematical depth involved in fully understanding normal distribution properties but also offers a valuable reference that could bridge gaps in existing literature.

Implications and Future Directions

The paper's contribution lies not in the originality of the formulas but in the clarity and completeness of their presentation. This consolidated resource has potential applications in statistical theory, enhancing methodologies for statistical inference and probabilistic analysis. In a world increasingly dominated by data-driven decision-making, having ready access to these statistical tools can significantly streamline complex analysis, whether in econometrics, psychometrics, or machine learning.

In practical terms, the availability of such formulas aids in the development of simulation studies and statistical software where precise modeling of distributions is necessary. The theoretical implications extend to educational contexts, where these derivations could serve as instructional material for advanced statistics courses.

Speculatively, as machine learning and AI continue to intersect with statistical physics and other sciences, understanding and utilizing distributions' higher-order moments could yield more nuanced models and insights, particularly in areas like deep learning where the sheer complexity of models often necessitates a statistical understanding of their underlying mechanisms.

In conclusion, "Moments and Absolute Moments of the Normal Distribution" serves as a robust technical summary of crucial statistical tools essential for both theoretical and applied statistical research.