{"paper":{"title":"Closed form expression of the multivariate standard Normal distribution under a weighted sum constraint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fr\\'ed\\'eric Vrins","submitted_at":"2018-01-19T12:59:49Z","abstract_excerpt":"In this letter we derive the $(n-1)$-dimensional distribution corresponding to a $n$-dimensional i.i.d. Normal standard vector $Z=(Z_1,Z_2,\\ldots,Z_n)$ subjected to the weighted sum constraint $\\sum_{i=1}^n w_i Z_i=c$, $w_i\\neq 0$. We first address the $n=2$ case before proceeding with the general $n\\geq 2$ case. The resulting distribution is a Normal distribution whose mean vector $\\mu$ and covariance matrix $\\Sigma$ are explicitly derived as a function of $w_1,\\ldots,w_n,c$. The derivation of the density relies on a very specific positive definite matrix for which the determinant and inverse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.06387","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"}