Antithetic Cauchy-tail integration on a finite interval ($\hat{p}=\frac{1}{2}-\frac{k}{2r}{\sum }_{j=1}^r[f(k{U}_j)+f(k(1-U_j))]$)

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.