Moments and Absolute Moments of the Normal Distribution

TL;DR

This paper compiles formulas for the raw and central moments and absolute moments of the normal distribution for real $ν > -1$, including both $E[X^ν]$ and $E[(X-μ)^ν]$ as well as $E[|X|^ν]$ and $E[|X-μ|^ν]$. The results are presented in closed form using special functions, notably the parabolic cylinder function $D_ν$, Kummer's confluent hypergeometric function $Φ$, Tricomi's function $Ψ$, and Gamma functions, with alternative representations where relevant. The central moments recover familiar parity results (nonzero only for even $ν$ with $(ν-1)!!$ when even) and the central absolute moments are μ-independent, while raw and absolute moments are expressed via hypergeometric or $D_ν$ forms. Derivations are provided using two integral identities involving $D_ν$ and Gamma integrals, linking Gaussian integrals to these special functions. The paper serves as a compact reference for moment formulas that are often scattered across textbooks, with explicit derivations and practical forms for applications in statistics and signal processing.

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.