{"paper":{"title":"On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Rajeshwari Majumdar, Suman Majumdar","submitted_at":"2017-10-24T05:30:16Z","abstract_excerpt":"We show that the orthogonal projection operator onto the range of the adjoint of a linear operator $T$ can be represented as $UT,$ where $U$ is an invertible linear operator. Using this representation we obtain a decomposition of a Normal random vector $Y$ as the sum of a linear transformation of $Y$ that is independent of $TY$ and an affine transformation of $TY$. We then use this decomposition to prove that the conditional distribution of a Normal random vector $Y$ given a linear transformation $\\mathcal{T}Y$ is again a multivariate Normal distribution. This result is equivalent to the well-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09285","kind":"arxiv","version":2},"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"}