{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"omni-math","formal_name":"Omni-MATH","introduction":"Omni-MATH evaluates mathematical reasoning on Olympiad-level problems. Its official dataset contains 4,428 problems accompanied by domain and difficulty information.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/KbsdJames/Omni-MATH","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"1d7ee4ba-9f4a-5391-80d3-1c1d67fbb1f1","task_key":"test--1d7ee4ba-9f4a-5391-80d3-1c1d67fbb1f1","task_revision_id":"1","upstream_id":"","short_description":"Assume that \\(X_{1}, \\ldots, X_{n} \\sim U[0,1]\\) (uniform distribution) are…","config":"","split":"test","body":"{\"problem\":\"Assume that \\\\(X_{1}, \\\\ldots, X_{n} \\\\sim U[0,1]\\\\) (uniform distribution) are i.i.d. Denote \\\\(X_{(1)}=\\\\min _{1 \\\\leq k \\\\leq n} X_{k}\\\\) and \\\\(X_{(n)}=\\\\max _{1 \\\\leq k \\\\leq n} X_{k}\\\\). Let \\\\(R=X_{(n)}-X_{(1)}\\\\) be the sample range and \\\\(V=\\\\left(X_{(1)}+X_{(n)}\\\\right) / 2\\\\) be the sample midvalue. (1). Find the joint density of \\\\(\\\\left(X_{(1)}, X_{(n)}\\\\right)\\\\). (2). Find the joint density of \\\\((R, V)\\\\). (3). Find the density of \\\\(R\\\\) and the density of \\\\(V\\\\).\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/KbsdJames/Omni-MATH","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}