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Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.

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Let $z_1 = x_1 - x_2$, and $z_2 = x_3 - x_4$, then $z_1, z_2 \sim \mathcal{N}(0,2\sigma^2)$.

Let $z = z_1 / z_2$. By results about ratios of Normal variates $z \sim \textrm{Cauchy}()$, i.e. is a standard Cauchy random variable.

Let $u = \frac{1}{\pi} \textrm{arctan}(z) + \frac{1}{2}$, the CDF of the standard Cauchy. Therefore $u \sim \textrm{U}([0,1])$.

Let $y = \sqrt{2} \,\textrm{erf}^{-1}(2u - 1)$, using the quantile function of the standard Normal.

We have $y \sim \mathcal{N}(0,1)$.

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    $\begingroup$ You can even make do with only $x_1, x_2, x_3$, since $(x_1-x_2)/(x_1-x_3)$ already has a non-trivial distribution which does not depend on $\mu$ and $\sigma$. $\endgroup$ – James Martin Dec 19 '18 at 11:19

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