{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-markov-inequality",
  "semantic_family_id": "sf-markov-inequality",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Markov不等式"
  },
  "summary": {
    "zh-Hant": "非負且有限期望的隨機變數在正閾值上的尾機率不超過E[X]/a。"
  },
  "statement": {
    "zh-Hant": "要求X>=0 almost surely、a>0及E[X]<infinity，結論P(X>=a)<=E[X]/a。完整有限有理law可精確算實際tail與bound；bound不等於tail，signed variable不可直接套用，零閾值禁止。"
  },
  "formula": {
    "tex": "M(X,a)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        },
        {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "cd070815ba10db5dbfe8da347e5f6e25070c0a3fb7d511dc8fbfc7619bca90ae"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整正規化有限有理law與total random-variable values。"
    },
    {
      "id": "a2",
      "zh-Hant": "主定理題的values全非負且threshold為正。"
    }
  ],
  "symbols": [
    {
      "token": "X",
      "role": "nonnegative_random_variable",
      "role_zh": "非負隨機變數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "positive_threshold",
      "role_zh": "正閾值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "X及其law。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "有限模型的期望值",
      "reason_zh": "E[X]。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "tail probability。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "偏序與全序",
      "reason_zh": "非負與threshold event。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "非負性、正閾值與有限期望是明示hypotheses。",
      "實際tail由完整law加總。",
      "Markov給upper bound而非equality。"
    ],
    "common_misconception_zh": "對可取負值X直接套Markov，或把E[X]/a當實際機率。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪exact題限完整有限law；一般測度版本只敘述不作有限抽樣證明。",
    "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Chebyshev inequality。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-markov-inequality.md 整理；2026-10-14 脈衝 R04（使用者手動接續 2026-11-07 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"
  }
}