{"paper":{"title":"Approximating Mills ratio","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Armengol Gasull, Frederic Utzet","submitted_at":"2013-07-12T12:23:56Z","abstract_excerpt":"Consider the Mills ratio $f(x)=\\big(1-\\Phi(x)\\big)/\\phi(x), \\, x\\ge 0$, where $\\phi$ is the density function of the standard Gaussian law and $\\Phi$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the whole $[0,\\infty)$ which allows to prove interesting properties where $f$ is involved. As applications we present a new proof that $1/f$ is strictly convex, and we give new sharp bounds of $f$ involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian $Q$--function are studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3433","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}