Upper and lower bounds for the reliability measure of a discrete distribution conditionally on the first three moments

TL;DR

This paper derives precise bounds for the reliability measure of a discrete random variable based on its first three moments, extending previous results to provide sharper estimates.

Contribution

It introduces new sharp bounds for the reliability measure of discrete distributions conditioned on the first three moments, advancing prior work in the field.

Findings

Established tight bounds for the reliability measure

Extended previous moment-based bounds to include the third moment

Provided mathematical proofs for the sharpness of bounds

Abstract

We give sharp bounds for the reliability measure of a discrete r.v. defined on {0, ..., n}, conditionally on the knowledge of the first three moments of the r.v. The present work is as an extension of the results given in [Di Cecco, Stat. Prob. Lett., 81(2011), 411-416].