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The quantile function (inverse CDF) for $X \\sim \\mbox{Cauchy}(0, 2)$ is $F^{-1}(x) = 2\\tan(\\pi(x-\\frac{1}{2}))$

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Given a random sample of (0.611, 0.358, 1.720, -4.556, 9.602) what are the order statistics and their associated approximate expected values?

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(Round to 3 decimal places)

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The first order approximation for $E[X_{(r)}] is F^{-1}\\Big[\\frac{r}{n+1}\\Big]$.

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Here we have $n=5$ and so once we identify the correct order for the random sample we just need to use $x=\\frac{r}{6}$ into the quantile function given.

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$r$$x_{(r)}$$\\approx E[X_{(r)}]$
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