{"paper":{"title":"On the expected time a branching process has K individuals alive","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Peter Neal, Tom Britton","submitted_at":"2013-04-30T14:41:26Z","abstract_excerpt":"Consider a homogeneous time-continuous branching process where individuals have constant birth rate $\\delta$, and life length distribution $Q$ having mean $E(Q)=1$. Let $X(u)$ denote the number of individuals alive at time $u$, and assume that $X(0)=1$. Let $K$ be a positive integer and define $A_K:=\\int_0^\\infty 1_{\\{X(u)=K\\}}du$, the accumulated time that the branching process has exactly $K$ individuals alive. In this paper we prove that $E(A_K)=\\delta^{K-1}/\\left(k(1\\vee\\delta)^K\\right)$, irrespective of the life length distribution $Q$, subject to the normalizing condition $E(Q)=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.8014","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"}