Moments and Absolute Moments of the Normal Distribution

I Introduction

The remainder of this text is structured as follows: Section II deals with preliminaries and introduces notation, particularly regarding some special functions. In Section III we present the results; the corresponding derivations are given in Section IV.

II Preliminaries

Kummer’s confluent hypergeometric functions:

Tricomi’s confluent hypergeometric functions:

III Results

In this section we give formulas for the raw/central (absolute) moments of a normal RV. If not noted otherwise, these results hold for $\nu>-1$.

IV Derivations

Central moments (17): Follows directly from (9) with $\Phi\!\left(\alpha,\gamma;0\right)=1$ and, hence,

To obtain (18) from (17) we use the identity [2, Sec. 8.334]

Then (19) follows from (18) by noting that

where we have used Kummer’s transformation [2, Sec. 9.212], i.e.,

Central absolute moments (24): Follows directly from (23) with $\Phi\!\left(\alpha,\gamma;0\right)=1$.

References