{
  "$schema": "../../../schemas/mko-v0.5.schema.json",
  "schema_version": "mko-v0.5",
  "id": "mko-standard-normal-distribution",
  "semantic_family_id": "sf-standard-normal-distribution",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "標準常態分布"
  },
  "summary": {
    "zh-Hant": "標準常態Z以Gaussian density phi(x)=(2pi)^(-1/2)exp(-x^2/2)與其連續CDF Phi定義，並具有mean 0、variance 1。"
  },
  "statement": {
    "zh-Hant": "本節點把Gaussian integral normalization、E[Z]=0與Var(Z)=1明列為named theorem facts，不以有限quadrature偽造完整推導。題庫可精確處理density symbolic form、evenness、phi(x)/phi(0)=exp(-x^2/2)，以及由declared normalization與symmetry推出Phi(0)=1/2與Phi(-x)=1-Phi(x)。一般nonzero x的CDF沒有elementary exact value；finite numerical integration不證normalization或moments，continuous CDF也不自動有density或是normal law。"
  },
  "formula": {
    "tex": "F(t)=P(Z\\le t)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>F</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>P</mi><mo>(</mo><mrow><mi>Z</mi><mo>≤</mo><mi>t</mi></mrow><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "t",
            "glyph": "t"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "P",
          "glyph": "P"
        },
        "arguments": [
          {
            "type": "less_equal",
            "left": {
              "type": "symbol",
              "name": "Z",
              "glyph": "Z"
            },
            "right": {
              "type": "symbol",
              "name": "t",
              "glyph": "t"
            }
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-source13",
      "version": "0.5.0",
      "source_sha256": "9201f8ed1b0f30ca60ffa7738020ba6dfe56a5513ef397c410bcab1acd926f68"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "符號題固定standard normal，不引入mu或sigma參數。"
    },
    {
      "id": "a2",
      "zh-Hant": "normalization與moments以明示named facts標記來源責任，非本輪有限計算結果。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "underlying_probability_measure",
      "role_zh": "底層機率空間的機率測度",
      "scope": "representative_formula"
    },
    {
      "token": "Z",
      "role": "standard_normal_random_variable",
      "role_zh": "標準常態隨機變數",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "continuous_cdf_with_gaussian_density",
      "role_zh": "標準常態累積分布函數",
      "scope": "representative_formula"
    },
    {
      "token": "t",
      "role": "real_cdf_argument",
      "role_zh": "累積分布函數的實數自變數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-continuous-real-distribution-function",
      "title_zh": "連續實值分布函數",
      "reason_zh": "continuous CDF。"
    },
    {
      "id": "mko-improper-integral",
      "title_zh": "瑕積分",
      "reason_zh": "density normalization背景。"
    },
    {
      "id": "mko-real-exponential-function",
      "title_zh": "實指數函數",
      "reason_zh": "Gaussian exponential。"
    },
    {
      "id": "mko-power-root",
      "title_zh": "冪與根式",
      "reason_zh": "normalization constant。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "有限分布的期望值",
      "reason_zh": "mean語義背景。"
    },
    {
      "id": "mko-variance",
      "title_zh": "有限機率分布的變異數",
      "reason_zh": "variance語義背景。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "density qualification與normalization theorem fact分開。",
      "evenness給對稱CDF identities。",
      "normalization與moments不由finite quadrature接受。",
      "記號對齊候選：P 是底層機率空間上的機率測度，P(Z≤t) 是可測事件 {ω:Z(ω)≤t} 的機率；F(t) 是標準常態 Z 的 CDF 值，不是把 Z 當事件或把隨機的 F(Z) 當標量 CDF 自變數。Gaussian normalization 與 moments 的既有 named-fact 責任不變。"
    ],
    "common_misconception_zh": "以幾個數值積分點宣稱Gaussian integral、mean或variance已證，或把任意continuous CDF稱normal。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不完整推導Gaussian integral、exp differentiation或integration by parts；這些保持named theorem facts。",
    "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": "加入有限變異數IID中央極限定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-standard-normal-distribution.md 整理；2026-10-15 脈衝 R04（使用者手動接續 2026-11-14 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}