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Given a PDF, can Mathematica tell me the name of a distribution? The obvious

ProbabilityDistribution[Exp[-x], {x, 0, ∞}]

does not evaluate usefully.

What I'm really interested in is

ProbabilityDistribution[(2 x)/(1 + x^2)^2, {x, 0, ∞}]

which is the ratio of two RayleighDistribution variables?

If I know the name for a distribution, an Internet search may reveal some relevant properties of the distribution.

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