{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-variance",
  "semantic_family_id": "sf-variance",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限機率模型的變異數"
  },
  "summary": {
    "zh-Hant": "本輪變異數為 E[(X−E[X])²]，不是 n−1 分母的樣本變異數；零權重離群值不影響它。"
  },
  "statement": {
    "zh-Hant": "令 μ=E[X]，對有限合法機率模型定義 V(X)=Var(X)=Σ_s w_s(X(s)−μ)²≥0。平移不改變變異數，Var(kX)=k²Var(X)；Var(X)=0 當且僅當 X 在所有正權重的基本結果上取同一個值。"
  },
  "formula": {
    "tex": "V(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>V</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "V",
        "glyph": "V"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "7d95e0a02a7f8e7156cbce75f3fbfedfa1f8b30614dc006f2890a160280d64a5"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限非空樣本空間、非負歸一化權重及實值總函數已明示。"
    },
    {
      "id": "a2",
      "zh-Hant": "使用機率分布變異數而非樣本估計；正權重 support 與全樣本空間須分開。"
    }
  ],
  "symbols": [
    {
      "token": "V",
      "role": "probability_variance",
      "role_zh": "有限機率分布的變異數",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "real_random_variable",
      "role_zh": "實值隨機變數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-expectation",
      "title_zh": "有限模型的期望值",
      "reason_zh": "變異數需先求期望。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "自然數冪與主平方根",
      "reason_zh": "偏差平方與 k² 的語義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每個加權平方項非負；零權重項可以有不同值，仍不影響矩數。零變異數指幾乎處處常值，在本輪有限模型就是正權重支持上常值。",
      "對 X+c 逐點平移後，平均也增加 c，偏差不變；對 kX 偏差乘 k，平方帶來 k²。k=0、負 k、單點支持都需保留。",
      "本輪精確有理矩數不使用浮點差 E[X²]−E[X]² 的數值近似，不套 n−1 樣本分母，也不把 Var(X+Y) 無條件分拆成兩個變異數。標準差及一般協方差另有義務。"
    ],
    "common_misconception_zh": "要求零變異數在零機率結果也同值，或混用樣本變異數／標準差。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限中心矩數可精確重算；一般方差、幾乎處處等價及可積性尚未附形式 Evidence。",
    "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": "綁定有限 centered moment、正支持常值與 ProbabilityTheory.variance 的橋接。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-variance.md 整理；2026-10-04 脈衝 R07（使用者手動接續 2026-10-05 工作序列；實際日期仍為 2026-10-04），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-04"
  }
}