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I have the following CDF:

 F[x_] = Piecewise[{
     {0, x < 0},
     {0.1 x^2, 0 <= x <= 2},
     {0.4, 2 < x <= 5/2},
     {0.5, 5/2 < x <= 4},
     {-0.5 E^-(x - 4)^2 + 1, 4 < x}
    }];

enter image description here

Now I define the a probability distribution as follows:

 dist = ProbabilityDistribution[{"CDF", F[x]}, {x, 0, Infinity}];
 G[x_] = CDF[dist, x];

But the plot of G now looks different:

enter image description here

It seems I lose the discontinuity point and on top of that the CDF G doesn't even go up to 1 anymore.


Question: Why do I lose the discontinuity of my CDF, and is there a way to avoid this issue?

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It appears that Mathematica does not support discontinuous CDFs. For example, try

F[x_] := ((Sign[x] + x) + 2)/4
dist = ProbabilityDistribution[{"CDF", F[x]}, {x, -1, 1}];
Plot[{F[x], CDF[dist, x]}, {x, -1, 1}]

and you can see the results are not the same. Even if we define $F$ equivalently as

G[x_] := Piecewise[{{(1 + x)/4, -1 <= x < 0}, {(3 + x)/4, 0 <= x <= 1}}]

we still don't get the right result.

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