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Related Experiment Video

Updated: Jun 11, 2026

Pupillometry to Assess Auditory Sensation in Guinea Pigs

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Published on: January 6, 2023

Bandwidth of gamma-distribution-shaped functions via Lambert W function.

Anthony LoPrete¹, Johannes Burge^1,2,3

¹Bioengineering Graduate Group, University of Pennsylvania, Philadelphia, PA, USA.

Statistics & Probability Letters

|June 10, 2026

Summary

Researchers derived a closed-form expression for the full width at half maximum (FWHM) of gamma-shaped functions. This provides a new analytic tool for characterizing bandwidth in scientific applications.

Keywords:

Full-width at half maximum Probability theory Special functions primary 60E05 secondary 33E20

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Area of Science:

Mathematics
Statistics
Signal Processing

Background:

The full width at half maximum (FWHM) is crucial for characterizing unimodal function bandwidth.
A closed-form expression for the FWHM of gamma-shaped functions is not readily available.

Purpose of the Study:

Derive and present a closed-form expression for the FWHM of gamma-shaped functions.
Provide an analytic tool for bandwidth characterization.

Main Methods:

Utilized the Lambert W function to compute the inverse of the gamma probability density function (PDF).
Derived an exact analytic expression for the full-width at an arbitrary y-proportion of the maximum (FWyM) of a gamma distribution.
Obtained the FWHM trivially from the FWyM expression.

Main Results:

Presented a novel, exact analytic expression for the FWHM of gamma-shaped functions.
Derived an expression for the octave bandwidth of gamma-shaped functions.
Compared the FWHM of gamma-shaped functions to their Gaussian approximation.

Conclusions:

The derived expression offers a valuable method for analyzing the bandwidth of gamma-shaped functions.
The study provides a practical MATLAB function for computing key quantities.
This work facilitates a deeper understanding and application of gamma-shaped functions in various scientific fields.