Handle Discontinuities in a CDF

Define a formula distribution by a cumulative distribution function. The distribution function contains jump discontinuities, which represent a mixture of continuous and discrete components.

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cdf = CDF[ MixtureDistribution[{1/3, 2/3}, {LaplaceDistribution[0, 1], TransformedDistribution[x - 2, x \[Distributed] BinomialDistribution[4, 1/3]]}], z];

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Plot[cdf, {z, -5, 5}, PlotTheme -> "Detailed", Filling -> Axis, ImageSize -> Medium, PlotLegends -> None]

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ProbabilityDistribution decomposes the distribution into absolutely continuous and discrete parts.

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ProbabilityDistribution[{CDF, cdf}, {z, -Infinity, Infinity}]

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Handle PDF input with DiracDelta weights.

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ProbabilityDistribution[ Sum[1/7 DiracDelta[x - k], {k, -3, 3}], {x, -Infinity, Infinity}]

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