{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-cumulative-distribution-function",
  "semantic_family_id": "sf-cumulative-distribution-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "累積分布函數"
  },
  "summary": {
    "zh-Hant": "實值隨機變數的CDF以F(t)=P(X小於等於t)定義，為單調右連續且有限離散跳躍等於點質量。"
  },
  "statement": {
    "zh-Hant": "完整有限實值分布下精確聚合所有x<=t的質量。CDF不是PMF或density；在support點的右值包含該點，jump為P(X=t)，端點極限為0與1。"
  },
  "formula": {
    "tex": "F(t)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>F</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "F",
        "glyph": "F"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "t",
          "glyph": "t"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8e63b590916a74c390ac2ecc4804e6fc24a7227c2db9165483e450450d0d60fa"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "互異整數support與精確有理非負質量總和1。"
    },
    {
      "id": "a2",
      "zh-Hant": "比較保留<=；所有輸出保持精確有理。"
    }
  ],
  "symbols": [
    {
      "token": "F",
      "role": "cumulative_distribution_function",
      "role_zh": "CDF",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "real_threshold",
      "role_zh": "實數門檻",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-discrete-probability-distribution",
      "title_zh": "有限離散機率分布",
      "reason_zh": "CDF由完整PMF聚合。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "序關係",
      "reason_zh": "事件使用X<=t。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "CDF值為事件機率。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "F為實數到0至1的函數。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "門檻包含等號。",
      "離散jump恰為點質量。",
      "單調與右連續由累積事件結構給出。"
    ],
    "common_misconception_zh": "把CDF值當單點PMF、漏掉等號，或用左連續取代右連續。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整有限離散law；不宣稱density或連續分布。",
    "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": "加入依分布收斂。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-cumulative-distribution-function.md 整理；2026-10-14 脈衝 R06（使用者手動接續 2026-10-30 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"
  }
}