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BenktanderGibratDistribution[a,b]

表示参数为 a 和 b 的第一类 Benktander 分布.

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背景
范例  
基本范例  
范围  
应用  
属性和关系  
巧妙范例  
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BenktanderGibratDistribution

BenktanderGibratDistribution[a,b]

表示参数为 a 和 b 的第一类 Benktander 分布.

更多信息

背景

范例

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基本范例  (4)

概率密度函数：

Wolfram Language code: Plot[Table[PDF[BenktanderGibratDistribution[a, 3], x], {a, {3, 4, 5}}]//Evaluate, {x, 1, 1.8}, Filling -> Axis]
Wolfram Language code: Plot[Table[PDF[BenktanderGibratDistribution[2, b], x], {b, {.5, 1, 2}}]//Evaluate, {x, 1, 2.2}, Filling -> Axis]
Wolfram Language code: PDF[BenktanderGibratDistribution[a, b], x]

累积分布函数：

Wolfram Language code: Plot[Table[CDF[BenktanderGibratDistribution[a, 3], x], {a, {3, 4, 5}}]//Evaluate, {x, 1, 1.8}, Filling -> Axis]
Wolfram Language code: Plot[Table[CDF[BenktanderGibratDistribution[2, b], x], {b, {.5, 1, 2}}]//Evaluate, {x, 1, 2.}, Filling -> Axis]
Wolfram Language code: CDF[BenktanderGibratDistribution[a, b], x]

均值和方差：

Wolfram Language code: Mean[BenktanderGibratDistribution[a, b]]
Wolfram Language code: Variance[BenktanderGibratDistribution[a, b]]//Together

中位数可以通过数值计算求得：

Wolfram Language code: Median[BenktanderGibratDistribution[2, 3]]//N

范围  (8)

生成一组服从 Benktander–Gibrat 分布的伪随机数：

Wolfram Language code: data = RandomVariate[BenktanderGibratDistribution[3, 3], 10 ^ 4];

比较其直方图与概率密度函数：

Wolfram Language code: Show[Histogram[data, {1, 2.5, .1}, "PDF"], Plot[PDF[BenktanderGibratDistribution[3, 3], x], {x, 1, 2.5}, PlotStyle -> Thick]]

分布参数估计：

Wolfram Language code: sample = RandomVariate[BenktanderGibratDistribution[3.5, 4], 10 ^ 3];

从样本数据估计分布参数：

Wolfram Language code: edist = EstimatedDistribution[sample, BenktanderGibratDistribution[a, b]]

比较样本的密度直方图和估计分布的概率密度函数：

Wolfram Language code: Show[Histogram[sample, {1, 2.5, 0.1}, "PDF"], Plot[PDF[edist, x], {x, 1, 2.5}, PlotStyle -> Thick]]

偏度：

Wolfram Language code: Plot3D[Skewness[BenktanderGibratDistribution[a, b]], {a, 3, 10}, {b, 1, 3}, MeshFunctions -> {#3&}, MeshShading -> ColorData[35, "ColorList"], AxesLabel -> Automatic]
Wolfram Language code: Skewness[BenktanderGibratDistribution[a, b]]

峰度：

Wolfram Language code: Plot3D[Kurtosis[BenktanderGibratDistribution[a, b]], {a, 3, 10}, {b, 1, 3}, MeshFunctions -> {#3&}, MeshShading -> ColorData[35, "ColorList"], AxesLabel -> Automatic]
Wolfram Language code: Kurtosis[BenktanderGibratDistribution[a, b]]

以参数的函数形式表示不同矩的解析式：

Wolfram Language code: FormulaGrid[list_, type_] := Grid[...]

Moment:

Wolfram Language code: FormulaGrid[Table[Moment[BenktanderGibratDistribution[a, b], r]//Together, {r, 3}], M]

具有符号式阶数的解析式：

Wolfram Language code: Moment[BenktanderGibratDistribution[a, b], r]

CentralMoment:

Wolfram Language code: FormulaGrid[Table[CentralMoment[BenktanderGibratDistribution[a, b], r]//FullSimplify, {r, 3}], CM]

FactorialMoment:

Wolfram Language code: FormulaGrid[Table[FactorialMoment[BenktanderGibratDistribution[a, b], r]//Together, {r, 3}], FM]

Cumulant:

Wolfram Language code: FormulaGrid[Table[Cumulant[BenktanderGibratDistribution[a, b], r]//FullSimplify, {r, 3}], C]

风险函数：

Wolfram Language code: Plot[Table[HazardFunction[BenktanderGibratDistribution[2, b], x], {b, {0.5, 1, 2}}]//Evaluate, {x, 1, 6}, Filling -> Axis, AxesOrigin -> {1, 0}]
Wolfram Language code: Plot[Table[HazardFunction[BenktanderGibratDistribution[a, 0.3], x], {a, {0.5, 1, 2}}]//Evaluate, {x, 1, 6}, Filling -> Axis, AxesOrigin -> {1, 0}]
Wolfram Language code: HazardFunction[BenktanderGibratDistribution[a, b], x]

分位数函数：

Wolfram Language code: Plot[Table[Quantile[BenktanderGibratDistribution[3, b], q], {b, {0.01, 1, 4}}]//Evaluate, {q, 0, 0.99}, Filling -> Axis]
Wolfram Language code: Plot[Table[Quantile[BenktanderGibratDistribution[a, 0.3], q], {a, {0.5, 1, 2}}]//Evaluate, {q, 0, 0.98}, Filling -> Axis]

使用无量纲的 Quantity 来定义 BenktanderGibratDistribution：

Wolfram Language code: 𝒟 = BenktanderGibratDistribution[Quantity[2*1, "Dozens"], 2.5]

应用  (3)

保险公司发现其索赔量（以免赔额为单位）服从 Benktander 第一类分布，其中参数 和 . 求索赔量大于2的概率：

Wolfram Language code: Probability[s > 2, sBenktanderGibratDistribution[1, 0.5]]

计算 Benktander 第一类分布的平均超限函数：

Wolfram Language code: Subscript[e, BI] = Expectation[x - sx > s, xBenktanderGibratDistribution[a, b], Assumptions -> s > 1]

对于较大的 ，它接近于对数正态分布，也称为 Gibrat 分布：

Wolfram Language code: Assuming[s > 0, Expectation[x - sx > s, xLogNormalDistribution[μ, σ]] == -s + (E^μ + (σ^2/2) Erfc[(Log[s] - μ - σ^2/Sqrt[2] σ)]/Erfc[(Log[s] - μ/Sqrt[2] σ)])//FullSimplify]

现在把余误差函数用它在较大自变量时的近似值代替：

Wolfram Language code: -s + (E^μ + (σ^2/2) Erfc[(Log[s] - μ - σ^2/Sqrt[2] σ)]/Erfc[(Log[s] - μ/Sqrt[2] σ)]) /. Erfc[x_] -> Normal[Series[Erfc[x], {x, ∞, 1}]]//FullSimplify

求与 Benktander 第一类分布相关联的稳态更新分布：

Wolfram Language code: StanionaryRenewalDistribution[dist_, lb_] := ProbabilityDistribution[{"CDF", (1 - Expectation[Max[x - t, 0], xdist] / (Mean[dist] - lb))}, {t, lb, Infinity}, Assumptions -> DistributionParameterAssumptions[dist]]
Wolfram Language code: srdist = StanionaryRenewalDistribution[BenktanderGibratDistribution[a, b], 1]//Simplify

与 BeniniDistribution 相比较：

Wolfram Language code: PDF[BeniniDistribution[a, b, 1], x]//Simplify

属性和关系  (3)

与其它分布的关系：

ParetoDistribution 是 BenktanderGibratDistribution 的极限情况：

Wolfram Language code: Limit[PDF[BenktanderGibratDistribution[a, b], x], b -> 0]
Wolfram Language code: PDF[ParetoDistribution[1, a + 1], x]
Wolfram Language code: % - %%

Benktander 第一类分布是次指数型的：

Wolfram Language code: dassum = DistributionParameterAssumptions[BenktanderGibratDistribution[a, b]];
Wolfram Language code: Limit[Exp[c x]Refine[PDF[BenktanderGibratDistribution[a, b], x], x > 1], x -> ∞, Assumptions -> c > 0 && dassum]

巧妙范例  (1)

取不同 α 值时的概率密度函数与累积分布函数等高图：

Wolfram Language code: dist = BenktanderGibratDistribution[a, 3];cdf = Function[{x, a}, Evaluate[CDF[dist, x]]]; ql = {0.025, 0.10, 0.25, 0.5, 0.75, 0.90, 0.975}; cl = Table[ColorData["Rainbow"][q], {q, Join[{0.0}, ql]}];
Wolfram Language code: Legended[Plot3D[PDF[dist, x], {x, 1, 2.2}, {a, 3, 7}, PlotTheme -> "Marketing", MeshFunctions -> {cdf}, Mesh -> {ql}, MeshStyle -> GrayLevel[0.8], PlotRange -> All, MeshShading -> cl, AxesLabel -> Automatic, BaseStyle -> Opacity[0.9], ImageSize -> 400], BarLegend["Rainbow", ql, LegendLabel -> "prob"]]
Wolfram Research (2010)，BenktanderGibratDistribution，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BenktanderGibratDistribution.html （更新于 2016 年）.

文本

Wolfram Research (2010)，BenktanderGibratDistribution，Wolfram 语言函数，https://reference.wolfram.com/language/ref/BenktanderGibratDistribution.html （更新于 2016 年）.

CMS

Wolfram 语言. 2010. "BenktanderGibratDistribution." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2016. https://reference.wolfram.com/language/ref/BenktanderGibratDistribution.html.

APA

Wolfram 语言. (2010). BenktanderGibratDistribution. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/BenktanderGibratDistribution.html 年

BibTeX

@misc{reference.wolfram_2026_benktandergibratdistribution, author="Wolfram Research", title="{BenktanderGibratDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/BenktanderGibratDistribution.html}", note=[Accessed: 04-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_benktandergibratdistribution, organization={Wolfram Research}, title={BenktanderGibratDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/BenktanderGibratDistribution.html}, note=[Accessed: 04-September-2026]}