RandomVariate of a custom distribution gives only negative points

Question

I don't understand why the following code

dis = ProbabilityDistribution[1/(x^2 + 1), {x, -5, 5}];
data = RandomVariate[dis, 10^4];;
Show[
 Histogram[data, 20, "ProbabilityDensity"],
 Plot[PDF[dis, x], {x, -5, 5}, PlotStyle -> Thick], 
 PlotRange -> {{-5, 5}, {0, 1}}]

Produces only negative data points? At the same time for the dis = NormalDistribution it works fine.

Answer 1

I understood it. The argument of the ProbabilityDistribution should be properly normalized. It would be actually useful if mathematica produced some kind of error message in this example. So the correct code should be:

norm = NIntegrate[1/(x^2 + 1), {x, -5, 5}];
dis = ProbabilityDistribution[(1/(x^2 + 1))/norm, {x, -5, 5}];
data = RandomVariate[dis, 10^4];
Show[
 Histogram[data, 20, "ProbabilityDensity"],
 Plot[PDF[dis, x], {x, -5, 5}, PlotStyle -> Thick], 
 PlotRange -> {{-5, 5}, {0, 0.4}}]
Export["c:/d/dist.png", %];