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Moments and Absolute Moments of the Normal Distribution
Andreas Winkelbauer
TL;DR
The notes address the incomplete textbook coverage of normal-distribution moment formulas by collecting formulas for raw, central, and absolute moments and giving their derivations. The formulas cover these moment types for ν > −1 and are derived using stated identities and special-function transformations.
Problem
Many textbooks omit at least some formulas for raw, central, and absolute moments of the normal distribution.
Method
The paper collects formulas for the normal distribution’s raw, central, raw absolute, and central absolute moments and derives them using identities and special-function transformations.
Results
The notes present formulas for raw, central, raw absolute, and central absolute moments of a normal random variable, generally for ν > −1.
Takeaways & Limitations
The notes provide a consolidated reference for normal-distribution moment formulas and their derivations.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The notes introduce a normal random variable and collect formulas for its raw, central, raw absolute, and central absolute moments. The formulas apply for real-valued ν > −1, with preliminaries, results, and derivations organized across subsequent sections.
- X is a normal random variable with mean µ = E{X} and variance σ2.
- The notes cover raw moments, central moments, raw absolute moments, and central absolute moments.
- The presented formulas hold for real-valued ν > −1.
- Section II introduces preliminaries and notation, Section III presents results, and Section IV gives their derivations.
II. PRELIMINARIES √
The preliminaries introduce notation and special functions used for the moment formulas and derivations.
- The preliminaries define subsequently used special functions.
- The notation includes nonnegative integers N0 = N∪{0}.
- The rising factorial is represented as z(z + 1) · · · (z + n − 1) for n ∈ N0.
- The double factorial expression z · (z − 2) · . . . · 3 · 1 is given for odd z ∈ N.
- The listed special-function families include Kummer’s and Tricomi’s confluent hypergeometric functions and parabolic cylinder functions.
III. RESULTS
This section presents formulas for the raw and central moments and absolute moments of a normal random variable, under the condition ν > −1.
- The results section gives formulas for raw, central, raw absolute, and central absolute moments of a normal random variable.
- The formulas hold for ν > −1 unless noted otherwise.
- Raw absolute moments are presented as a distinct results category.
- Central absolute moments are presented as a distinct results category.
IV. DERIVATIONS
The derivations section obtains the displayed moment formulas from integral identities, special-function identities, direct substitutions, and Kummer’s transformation.
- The derivations use two identities valid for γ ∈ R and ν > −1.
- The raw moments are derived using the stated identities.
- The central moments follow directly from the raw-moment formula with Φ(α, γ; 0) = 1.
- An additional identity is used to obtain the next central-moment expression, followed by a simplification for the subsequent formula.
- The raw absolute moments are derived separately using Kummer’s transformation in the final step.
- The central absolute moments follow directly from the raw absolute-moment formula with Φ(α, γ; 0) = 1.