Define a Distribution Given Its Hazard Function 

A hazard function is given.

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X h[x_] := Piecewise[{{1, x < 1}, {Exp[-x], x >= 1}}]

Define a ProbabilityDistribution using the hazard function.

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X \[ScriptCapitalD] = ProbabilityDistribution[{"HF", h[x]}, {x, 0, \[Infinity]}]

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Plot distribution functions.

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X GraphicsGrid[ Partition[ Table[Plot[fun[\[ScriptCapitalD], x], {x, 0, 4}, Filling -> Axis, PlotLabel -> fun, ImageSize -> 200], {fun, {PDF, CDF, HazardFunction, SurvivalFunction}}], 2, 2]]

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Simulate and plot the histogram.

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X data = RandomVariate[\[ScriptCapitalD], 10^5]; Histogram[data, {0, 6, .1}, "PDF", ImageSize -> 300]

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