{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-cumulative-distribution-function-01",
  "mko_id": "mko-cumulative-distribution-function",
  "title": {
    "zh-Hant": "三點分布的完整CDF",
    "en": "The complete CDF of a three-point law"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-cumulative-distribution-function-01",
    "source": {
      "mko_id": "mko-cumulative-distribution-function",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "4111d0774535349038ca5452c94102adeb111e4fa99147bfbaf9d380f06da4fc",
      "english_sha256": "18c0763baf083d096e31a7b58bbc126f1812a47b479a008f8982c620487c75d7"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "算出全實軸CDF、右連續值與跳躍。",
      "en": "Compute the CDF on the whole real line, including its right-continuous values and its jumps."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "X的support為{-1,1,3}，對應質量1/4、1/2、1/4。",
          "en": "X has support{-1,1,3} with masses 1/4,1/2,1/4."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "p=(-1:1/4,1:1/2,3:1/4)",
        "after_tex": "1/4+1/2+1/4=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個非負有理質量構成完整law。",
          "en": "The three nonnegative rational masses form a complete law."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "support互異，沒有重複質量須合併。",
            "en": "The support values are distinct; no duplicate mass is hidden."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "F(t)=P(X\\le t),\\ t<-1",
        "after_tex": "F(t)=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "門檻在最小support左側，累積事件為空。",
          "en": "Below the least support value the cumulative event is empty."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "t=-1不屬此區間，因定義包含等號。",
            "en": "The endpoint -1 is excluded because the CDF includes equality."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "-1\\le t<1",
        "after_tex": "\\{X\\le t\\}=\\{X=-1\\},\\ F(t)=1/4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "這一整段只含-1的質量，包括左端。",
          "en": "This whole interval includes only the mass at -1, including its left endpoint."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能在t=-1用左極限0代替函數值。",
            "en": "At t=-1 the left limit0 is not the function value."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "1\\le t<3",
        "after_tex": "F(t)=1/4+1/2=3/4",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "跨過1時加入其完整1/2質量。",
          "en": "Crossing1 adds its full mass 1/2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "CDF為3/4，不是單點P(X=1)=1/2。",
            "en": "The CDF is 3/4, not the point mass P(X=1)=1/2."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "t\\ge3",
        "after_tex": "F(t)=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "最後一個區間含所有三點。",
          "en": "The final interval includes all three points."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "t=3必須在此段而不是前段。",
            "en": "The point 3 belongs here, not in the preceding interval."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "F(-1)-F(-1^-),F(1)-F(1^-),F(3)-F(3^-)",
        "after_tex": "(1/4,1/2,1/4)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個jump分別回復原PMF。",
          "en": "The three jumps recover the original PMF."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "左極限依前一段，不能以右值相減。",
            "en": "Use the preceding interval for left limits, not another right value."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "0\\le1/4\\le3/4\\le1",
        "after_tex": "F\\text{ is nondecreasing and right-continuous}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "常數區間與左閉右開的接合使每個support點右連續。",
          "en": "Constant intervals joined left-closed and right-open give right continuity at each support point."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "跳躍允許不連續；右連續不等於雙側連續。",
            "en": "Jumps are allowed; right continuity is not two-sided continuity."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "t<-1\\text{ or }t\\ge3",
        "after_tex": "\\lim_{t\\to-\\infty}F(t)=0,\\quad\\lim_{t\\to\\infty}F(t)=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "左右尾為恆0與恆1，端點極限因此完成。",
          "en": "The tails are identically0 and 1, proving both endpoint limits."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "全實軸四段已覆蓋，非只測三個t值。",
            "en": "All four real-line regions are covered, not merely three test points."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "CDF四段依序為0、1/4、3/4、1，三個jump正好是原質量。",
      "en": "The four CDF regions have values 0,1/4,3/4,1; its jumps are precisely the masses."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "F(1)=1/2。",
          "en": "F(1)=1/2."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "還須包含-1的1/4質量，故為3/4。",
          "en": "The mass 1/4 at -1 is also included, giving 3/4."
        }
      },
      {
        "claim": {
          "zh-Hant": "F(-1)=0。",
          "en": "F(-1)=0."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "事件使用≤，因此包含-1。",
          "en": "The event uses≤ and includes-1."
        }
      },
      {
        "claim": {
          "zh-Hant": "此CDF在1連續。",
          "en": "This CDF is continuous at 1."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "左右差1/2，只有右連續。",
          "en": "Its jump is 1/2; it is only right-continuous there."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "For a complete finite real-valued law, aggregate exactly every mass with x≤t. The CDF is not a PMF or density. Its value at a support point includes that point's mass, its jump is P(X=t), and its endpoint limits are 0 and 1."
        },
        "source": {
          "mko_id": "mko-cumulative-distribution-function",
          "sha256": "4111d0774535349038ca5452c94102adeb111e4fa99147bfbaf9d380f06da4fc",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限law的事件機率為所含質量和；CDF定義F(t)=P(X≤t)，jump為F(t)-F(t-)。",
          "en": "Event probabilities sum included masses; F(t)=P(X≤t), with jump F(t)-F(t-)."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "1/4+1/2+1/4",
        "rhs": "1"
      },
      {
        "id": "N02",
        "lhs": "3/4-1/4",
        "rhs": "1/2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整有限離散分布；不把CDF當density。",
      "en": "A complete finite discrete law, not a density calculation."
    },
    "md_symbols": [
      [
        "F",
        "累積分布函數",
        "cumulative distribution function"
      ],
      [
        "t",
        "實數門檻",
        "real threshold"
      ]
    ],
    "md_error_controls": [
      {
        "title": "端點 / Endpoints",
        "zh": "四段使用左閉右開，最右段含3。",
        "en": "Use left-closed/right-open pieces and include3 in the last piece."
      },
      {
        "title": "CDF與PMF / CDF versus PMF",
        "zh": "用相鄰累積值之差還原點質量。",
        "en": "Recover point masses by differences of adjacent cumulative values."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
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    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}