Moments by Integrating the Moment-Generating Function

TL;DR

This paper presents a new method called CMGF for deriving a wide range of moments, including fractional and absolute moments, using the moment-generating function without derivatives, demonstrated through three applications.

Contribution

The paper introduces CMGF, a novel approach that simplifies the calculation of various moments directly from the MGF, including fractional and complex moments, without needing derivatives.

Findings

Effective for moments with closed-form MGFs

Simplifies fractional and complex moment calculations

Applicable when density functions are difficult to obtain

Abstract

We introduce a novel method for obtaining a wide variety of moments of any random variable with a well-defined moment-generating function (MGF). We derive new expressions for fractional moments and fractional absolute moments, both central and non-central moments. The expressions are relatively simple integrals that involve the MGF, but do not require its derivatives. We label the new method CMGF because it uses a complex extension of the MGF and can be used to obtain complex moments. We illustrate the new method with three applications where the MGF is available in closed-form, while the corresponding densities and the derivatives of the MGF are either unavailable or very difficult to obtain.