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Other

Degeneracy of the Cauchy-Distribution

The pdf of a (standard) Cauchy-Distribution is

$$p(x)=\frac{1}{\pi(1+x^2)}$$

Hence, the mean calculates as

$$\mathbb{E}\left[X\right]=\int_{-\infty}^{\infty}x\frac{1}{\pi(1+x^2)}dx=\frac{1}{2\pi}log(1+x^2)\Big|^\infty_{-\infty}=``\infty-\infty``$$

which is an undefined expression and the distribution does not have a mean. Any higher moments are also undefined which shows a limitation of methods using statistical moments.