Why does the Cauchy distribution have no mean?

hard

Answer

$1\; / \;({\pi}(1 + x^{2}))$
$1 / x^{2}$
$Consequence: sample mean of Cauchy does not converge (LLN fails).$
$Sums of Cauchy are still Cauchy (heavy-tail preservation), so CLT doesn't apply either.$
$Use case: robust statistics use it as a heavy-tailed model for outliers; ratio of two normals has Cauchy distribution.$

Check yourself — multiple choice

Mean is 0
Tails ∝ $1 / x^{2}$ → ∫x·f(x) diverges → no mean; LLN and CLT fail; ratio of normals is Cauchy
Same as normal
Has finite variance

Cauchy: undefined mean due to heavy tails; LLN/CLT don't apply.

#distributions#probability

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