Normalization of Formula Distributions

Use ProbabilityDistribution with option Method->"Normalize" to automatically compute the normalization coefficient.

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uv\[ScriptCapitalD] = ProbabilityDistribution[Sin[x] Exp[-x], {x, 0, Pi}, Method -> "Normalize"]

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Sample from the distribution and compare the histogram with the density function.

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Show[Histogram[RandomVariate[uv\[ScriptCapitalD], 10^4], Automatic, PDF, PlotTheme -> "Detailed"], Plot[PDF[uv\[ScriptCapitalD], x], {x, 0, Pi}, PlotTheme -> "Marketing"], ImageSize -> Medium]

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Normalize a multivariate distribution with option Method->"Normalize".

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mv\[ScriptCapitalD] = ProbabilityDistribution[Boole[x^2 + y^2 < 1], {x, 0, 1}, {y, 0, 1}, Method -> "Normalize"]

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Show[Histogram3D[ RandomVariate[mv\[ScriptCapitalD], 10^4, Method -> "AcceptanceRejection"], 10, PDF, PlotTheme -> "Detailed"], Plot3D[PDF[mv\[ScriptCapitalD], {x, y}], {x, 0, 1}, {y, 0, 1}, PlotStyle -> None], ImageSize -> Medium]

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