PDF for Product/Quotients of Random Variables

Find the probability density function for the ratio of the smallest to the largest sample among independent drawings from BetaDistribution[2, 3].

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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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Visualize the density.

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

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Compute the PDF for the product of two triangular distributions.

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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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Find the PDF for the quotient of two independent normal random variables.

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

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The distribution is heavy tailed for any fixed value of .

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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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