{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-joint-distribution",
  "semantic_family_id": "sf-finite-joint-distribution",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限聯合分布"
  },
  "summary": {
    "zh-Hant": "共同有限機率空間上多個隨機變數的值組與精確質量共同指定有限聯合分布。"
  },
  "statement": {
    "zh-Hant": "二維有限聯合分布列出互異值對(x,y)及非負精確質量，總和恰為1。邊際、矩形事件及單點質量都從同一joint law聚合。"
  },
  "formula": {
    "tex": "J(X,Y)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>J</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "J",
        "glyph": "J"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        },
        {
          "type": "symbol",
          "name": "Y",
          "glyph": "Y"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b66b6109ed2d2d8b2997f6a83e757d07bf4a5541136d2314b0fdb27c71b40b3f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "support為互異整數對，質量為精確有理數。"
    },
    {
      "id": "a2",
      "zh-Hant": "joint law是數學模型，不宣稱經驗資料配適或連續密度。"
    }
  ],
  "symbols": [
    {
      "token": "J",
      "role": "joint_distribution",
      "role_zh": "有限聯合分布",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable",
      "role_zh": "第一隨機變數",
      "scope": "representative_formula"
    },
    {
      "token": "Y",
      "role": "random_variable",
      "role_zh": "第二隨機變數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-discrete-probability-distribution",
      "title_zh": "有限離散機率分布",
      "reason_zh": "聯合值對形成有限support及PMF。"
    },
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "座標值來自共同樣本空間上的變數。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "事件質量依機率公理。"
    },
    {
      "id": "mko-sample-space",
      "title_zh": "樣本空間",
      "reason_zh": "共同機率空間是joint語義根基。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "兩個邊際由同一support分別聚合。",
      "矩形事件要求兩座標同時落入指定集合。",
      "邊際通常不能反推完整joint law。"
    ],
    "common_misconception_zh": "把兩個邊際表直接當聯合分布，或容許質量只近似和為1。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限joint PMF；不外推連續分布。",
    "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": "加入協方差、隨機變數獨立與條件期望。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-joint-distribution.md 整理；2026-10-13 脈衝 R04（使用者手動接續 2026-10-26 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"
  }
}