Editing Normal distribution (section)

===== Operations on two independent standard normal variables =====
If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two independent standard normal random variables with mean 0 and variance 1, then
* Their sum and difference is distributed normally with mean zero and variance two: <math display=inline>X_1 \pm X_2 \sim \mathcal{N}(0, 2)</math>.
* Their product <math display=inline>Z = X_1 X_2</math> follows the [[product distribution#Independent central-normal distributions|product distribution]]<ref>{{cite web|url = http://mathworld.wolfram.com/NormalProductDistribution.html |title = Normal Product Distribution|work = MathWorld |publisher =wolfram.com| first = Eric W. |last = Weisstein}}</ref> with density function <math display=inline>f_Z(z) = \pi^{-1} K_0(|z|)</math> where <math display=inline>K_0</math> is the [[Macdonald function|modified Bessel function of the second kind]]. This distribution is symmetric around zero, unbounded at <math display=inline>z = 0</math>, and has the [[characteristic function (probability theory)|characteristic function]] <math display=inline>\phi_Z(t) = (1 + t^2)^{-1/2}</math>.
* Their ratio follows the standard [[Cauchy distribution]]: <math display=inline>X_1/ X_2 \sim \operatorname{Cauchy}(0, 1)</math>.
* Their Euclidean norm <math display=inline>\sqrt{X_1^2 + X_2^2}</math> has the [[Rayleigh distribution]].