Expectation of CauchyDistribution

Question

Bug introduced in 8.0 or earlier and persisting through 13.2.0

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I've noticed this strange behavior and I'm wondering if it's a bug.

I define a Cauchy distribution:

c = CauchyDistribution[0, 1];

If I evaluate Mean[c], I get Indeterminate, as expected.

If I evaluate Expectation[x, x \[Distributed] c], I get Expectation[x, x \[Distributed] CauchyDistribution[0, 1]]. I would have expected Indeterminate, too, but that's ok.

However, if I evaluate Expectation[x + y, {x \[Distributed] c, y \[Distributed] c}], I get 0.

The sum of two independent Cauchy distributed random variables should be another Cauchy distributed random variable, right? Why is the expectation 0?

Answer 1

You can check that :

TransformedDistribution[ x + y, {x \[Distributed] CauchyDistribution[0, 1], 
                                y \[Distributed] CauchyDistribution[0, 1]}]

(* CauchyDistribution[0, 2] *)  

Expectation[z, z \[Distributed] CauchyDistribution[0, 2]]
(* Expectation[z, z \[Distributed] CauchyDistribution[0, 2]] *)

Mean[TransformedDistribution[ x + y, {x \[Distributed] c, y \[Distributed] c}]]
(* Indeterminate *)