A Simple Expression for Mill's Ratio of the Student's $t$-Distribution

TL;DR

This paper presents a simple formula for Mill's ratio of the Student's t-distribution and uses it to prove a conjecture related to finite-time analysis in multi-armed bandit problems.

Contribution

It introduces a straightforward expression for Mill's ratio and applies it to prove a conjecture in the analysis of multi-armed bandit algorithms.

Findings

Derived a simple expression for Mill's ratio of Student's t-distribution

Proved Conjecture 1 from Auer et al. (2002) using the new expression

Enhanced understanding of tail probabilities in statistical distributions

Abstract

I show a simple expression of the Mill's ratio of the Student's t-Distribution. I use it to prove Conjecture 1 in P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite-time analysis of the multiarmed bandit problem. Mach. Learn., 47(2-3):235--256, May 2002.