Wolfram 语言™

随机变量的积/商的 PDF

找出 BetaDistribution[2, 3]的 个独立抽样中最小与最大样本比值的概率密度函数.

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n = 5; pdf = PDF[ TransformedDistribution[ min/max, {min, max} \[Distributed] OrderDistribution[{BetaDistribution[2, 3], n}, {1, n}]], u]

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可视化密度.

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Plot[pdf, {u, 0, 1}, PlotRange -> All, Filling -> Axis, PlotTheme -> "Detailed", ImageSize -> Medium, PlotLegends -> None]

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计算两个三角形分布的乘积的 PDF.

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pdf2 = PDF[ TransformedDistribution[ x1 x2, {x1 \[Distributed] TriangularDistribution[{-1, 2}, -1], x2 \[Distributed] TriangularDistribution[{-4, 3}, 2]}], u]

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Plot[pdf2, {u, -4, 4}, Exclusions -> None, Filling -> Axis, PlotTheme -> "Detailed", ImageSize -> "Medium", PlotLegends -> None, PlotRange -> All]

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找出两个独立正态随机变量商的 PDF.

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pdf3 = PDF[ TransformedDistribution[ z1/z2, {z1 \[Distributed] NormalDistribution[], z2 \[Distributed] NormalDistribution[\[Mu], 1]}], x]

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任何 为固定值的分布都为重尾分布.

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Series[Exp[\[Mu]^2/2] pdf3, {x, Infinity, 8}, Assumptions -> \[Mu] > 0] // Expand

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Plot[Evaluate[ pdf3 /. {{\[Mu] -> 0}, {\[Mu] -> 1}, {\[Mu] -> 3}, {\[Mu] -> 5}}], {x, -2, 2}, PlotLegends -> {"\[Mu] = 0", "\[Mu] = 1", "\[Mu] = 3", "\[Mu] = 5"}, PlotRange -> All, PlotTheme -> "Detailed", ImageSize -> "Medium"]

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