# Existence or impossibility of Gaussian factory

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

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 $$y \sim \mathcal{N}(0,1)$$.
• 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 $\sigma$. – James Martin Dec 19 '18 at 11:19