TITLE: How to transform gaussian(normal) distribution to uniform distribution?
QUESTION [14 upvotes]: I have gaussian distributed numbers with mean 0 and variance 0.2.
And I want to transform this distribution to uniform distribution [-3 3].
How can I transform gaussian distribution numbers to uniform distribution?

REPLY [7 votes]: \begin{align}
X & \sim N(0,0.2) = N\left( 0, \frac 1 5 \right) \\[10pt]
X\cdot\sqrt 5 &\sim N(0,1) \\[10pt]
\Phi(X\cdot\sqrt 5) & \sim \operatorname{Uniform}[0,1] \tag 1 \\[10pt]
-3 + 6 \Phi(X\cdot \sqrt 5) & \sim \operatorname{Uniform}[-3,3] 
\end{align}
A proof of line $(1)$ is as follows: Suppose $Z\sim N(0,1).$ Then for $0 \le x\le 1,$
$$
\Pr(\Phi(Z) \le x) = \Pr( Z \le \Phi^{-1}(x)) = \Phi(\Phi^{-1}(x)) = x.
$$