[Paper Review] Moments and Absolute Moments of the Normal Distribution

Andreas Winkelbauer|arXiv (Cornell University)|Sep 19, 2012

Statistical Mechanics and EntropyPhysics and Astronomy1 references125 citations

TL;DR

This paper provides comprehensive formulas for raw and central moments, as well as absolute moments, of the normal distribution for real-valued $ u > -1$. It derives these using parabolic cylinder functions and confluent hypergeometric functions, offering closed-form expressions that generalize known results and fill gaps in standard textbooks.

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.

Motivation & Objective

• To systematically derive and compile formulas for raw, central, absolute, and central absolute moments of the normal distribution.
• To address the omission of such formulas in many standard textbooks, particularly for non-integer $ uf$.
• To present unified expressions using special functions like parabolic cylinder and confluent hypergeometric functions.
• To provide derivations based on integral identities and properties of special functions, ensuring mathematical rigor.
• To extend known results beyond integer-order moments to real $ uf > -1$, enhancing theoretical and applied utility.

Proposed method

• Derives raw moments using the characteristic function and integral identities involving $D_{ u}(z)$, the parabolic cylinder function.
• Expresses moments via Kummer’s confluent hypergeometric function $\Phi(\alpha,\gamma;z)$ and Tricomi’s function $\Psi(\alpha,\gamma;z)$.
• Applies the identity $\int_{-\infty}^{\infty}(-jx)^\nu e^{-x^2 + jx\gamma}dx = \sqrt{2^{-\nu}\pi}e^{-\gamma^2/8}D_\nu(\gamma/\sqrt{2})$ as a key integral tool.
• Uses the transformation $\Phi(\alpha,\gamma;z) = e^z\Phi(\gamma-\alpha,\gamma;-z)$ to simplify expressions involving $\exp(-\mu^2/(2\sigma^2))$.
• Applies the reflection formula $\Gamma(\frac{1+\nu}{2})\Gamma(\frac{1-\nu}{2}) = \frac{\pi}{\cos(\pi\nu/2)}$ to connect parabolic cylinder and trigonometric forms.
• Derives central absolute moments by setting $\mu=0$ in the raw absolute moment formula, leveraging $\Phi(\alpha,\gamma;0) = 1$.

Experimental results

Research questions

• RQ1 What are the closed-form expressions for the $\nu$-th raw and central moments of a normal random variable for real $\nu > -1$?
• RQ2 How can the absolute moments $\mathrm{E}[|X|^\nu]$ and $\mathrm{E}[|X-\mu|^\nu]$ be expressed in terms of special functions?
• RQ3 What is the connection between parabolic cylinder functions and confluent hypergeometric functions in expressing these moments?
• RQ4 How do the derived formulas reduce to known results (e.g., $(\nu-1)!!$ for even $\nu$) in the integer case?
• RQ5 Can the formulas be unified across different parameter regimes (e.g., $\mu \leq 0$ vs. $\mu > 0$) using special function identities?

Key findings

• The $\nu$-th raw moment of $X \sim \mathcal{N}(\mu,\sigma^2)$ is given by $\mathrm{E}[X^\nu] = (j\sigma)^\nu \exp(-\mu^2/(4\sigma^2)) D_\nu(-j\mu/\sigma)$, valid for $\nu > -1$.
• For central moments, $\mathrm{E}[(X-\mu)^\nu] = \sigma^\nu 2^{\nu/2 - 1} \frac{\Gamma((\nu+1)/2)}{\sqrt{\pi}} (1 + (-1)^\nu)$, yielding $0$ for odd $\nu$ and $(\nu-1)!!\sigma^\nu$ for even $\nu$.
• The raw absolute moment is $\mathrm{E}[|X|^\nu] = \sigma^\nu 2^{\nu/2} \frac{\Gamma((\nu+1)/2)}{\sqrt{\pi}} \Phi(-\nu/2, 1/2; -\mu^2/(2\sigma^2))$.
• The central absolute moment simplifies to $\mathrm{E}[|X-\mu|^\nu] = \sigma^\nu 2^{\nu/2} \frac{\Gamma((\nu+1)/2)}{\sqrt{\pi}}$, independent of $\mu$.
• When $\nu$ is a non-negative integer, the formulas reduce to standard results: even-order central moments are $\sigma^\nu (\nu-1)!!$, odd-order ones are zero.
• The use of $\Psi$ and $\Phi$ functions allows piecewise expressions for $\mathrm{E}[X^\nu]$ depending on the sign of $\mu$, with $\Psi^*$ for $\mu > 0$.