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

  • 1
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.