{"ID":2874353,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.19307","arxiv_id":"2509.19307","title":"Bandwidth of Gamma-Distribution-Shaped Functions via Lambert W Function","abstract":"The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma distribution probability density function (PDF)-is not widely available. Here, we derive and present just such an expression. To do so, we use the Lambert W function to compute the inverse of the gamma PDF. We use this inverse to derive an exact analytic expression for the width of a gamma distribution at an arbitrary proportion of the maximum, from which the FWHM follows trivially. (An expression for the octave bandwidth of gamma-shaped functions is also provided.) The FWHM is then compared to the Gaussian approximation of gamma-shaped functions. A few other related issues are discussed.","short_abstract":"The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma distribution probability density function (PDF)-is not widely available. Here, we deri...","url_abs":"https://arxiv.org/abs/2509.19307","url_pdf":"https://arxiv.org/pdf/2509.19307v1","authors":"[\"Anthony LoPrete\",\"Johannes Burge\"]","published":"2025-09-05T19:11:36Z","proceeding":"eess.SP","tasks":"[\"eess.SP\",\"math.PR\"]","methods":"[]","has_code":false}