{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-chebyshev-inequality",
  "semantic_family_id": "sf-chebyshev-inequality",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Chebyshev不等式"
  },
  "summary": {
    "zh-Hant": "有限均值與變異數的隨機變數偏離均值至少epsilon的機率不超過variance/epsilon²。"
  },
  "statement": {
    "zh-Hant": "epsilon>0且mean、variance有限。事件|X-mu|>=epsilon等同(X-mu)^2>=epsilon^2，對非負squared deviation套Markov。上界不等於actual tail，不需要independence；zero mass atoms與zero variance按a.s.語義處理。"
  },
  "formula": {
    "tex": "C(X,e)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        },
        {
          "type": "symbol",
          "name": "e",
          "glyph": "e"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5e4eaba08c1be9e448b9578711b898f6bceaea86e350624211eae53337d823ab"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整正規化有限有理law與total random variable。"
    },
    {
      "id": "a2",
      "zh-Hant": "epsilon為正有理數，mean與second moment exact。"
    }
  ],
  "symbols": [
    {
      "token": "X",
      "role": "finite_random_variable",
      "role_zh": "有限隨機變數",
      "scope": "representative_formula"
    },
    {
      "token": "e",
      "role": "positive_deviation",
      "role_zh": "正偏差閾值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-markov-inequality",
      "title_zh": "Markov不等式",
      "reason_zh": "squared-deviation derivation。"
    },
    {
      "id": "mko-variance",
      "title_zh": "有限機率模型的變異數",
      "reason_zh": "variance。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "deviation event。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "squared deviation是nonnegative。",
      "Markov threshold為epsilon squared。",
      "Chebyshev不要求independence且只給upper bound。"
    ],
    "common_misconception_zh": "把bound當tail equality，或令epsilon=0。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite law是exact instance，不是有限simulation證一般定理。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入total variance與更強concentration bounds。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-chebyshev-inequality.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-08 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}