= Antithetic Cauchy-tail integration on a finite interval
{title2=$\widehat p=\frac12-\frac{k}{2r}\sum_{j=1}^r[f(kU_j)+f(k(1-U_j))]$}
For standard <Cauchy distribution> <probability density function> $f$ and $k>0$, this estimates its upper-tail <probability> without sampling a heavy tail. The function $f(ku)$ decreases on $[0,1]$, while $f(k(1-u))$ increases, so their <covariance> is nonpositive. The paired estimator is unbiased and has no larger <variance> than an estimator using the same number $2r$ of <independent> <probability density function> evaluations.
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