{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-discrete-probability-distribution",
  "semantic_family_id": "sf-finite-discrete-probability-distribution",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限離散機率分布"
  },
  "summary": {
    "zh-Hant": "有限離散分布是在有限結果值上指定非負機率質量，全部質量精確和為一。"
  },
  "statement": {
    "zh-Hant": "有限離散隨機變數的機率質量函數p對每個有限值域元素指定非負機率，且所有質量和為1。事件機率由相應質量相加，期望為值與質量的加權和。"
  },
  "formula": {
    "tex": "p:X\\to R",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>R</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "p",
        "glyph": "p"
      },
      "domain": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      },
      "codomain": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "9ccd8b2931292a6db80198e8877f169de6681df2e2b299c26dcfe3480467221f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "結果值互異，質量使用精確有理數；合格PMF要求每項非負且總和恰為1。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫只表示數學分布，不宣稱來自樣本估計、資料擬合或真實族群。"
    }
  ],
  "symbols": [
    {
      "token": "p",
      "role": "probability_mass_function",
      "role_zh": "機率質量函數",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "finite_value_set",
      "role_zh": "有限值域",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "real_codomain",
      "role_zh": "實數陪域",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "分布描述隨機變數取值的機率法則。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "質量遵守非負與總和一。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "PMF為每個值指定唯一質量。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "PMF資格先核非負與總和一；不合格時不能繼續當機率分布求期望。",
      "事件機率是事件所含值的質量和；事件外值不加入。",
      "推前分布把映到同一新值的原質量相加；這仍是精確數學轉換，不是統計估計。"
    ],
    "common_misconception_zh": "只因每項介於0與1就忽略總和，或把理論PMF冒充資料擬合結果。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限精確PMF；不宣稱連續密度、估計品質或實證適配。",
    "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": "加入聯合分布、離散隨機過程與有限Markov鏈。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-discrete-probability-distribution.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-22 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}