Editing Normal distribution (section)

==== Operations on a single normal variable ====
If {{tmath|X}} is distributed normally with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, then
* <math display=inline>aX+b</math>, for any real numbers {{tmath|a}} and {{tmath|b}}, is also normally distributed, with mean <math display=inline>a\mu+b</math> and variance <math display=inline>a^2\sigma^2</math>. That is, the family of normal distributions is closed under [[linear transformations]].
* The exponential of {{tmath|X}} is distributed [[Log-normal distribution|log-normally]]: <math display=inline>e^X \sim \ln(N(\mu, \sigma^2))</math>.
* The standard [[logistic function|sigmoid]] of {{tmath|X}} is [[Logit-normal distribution|logit-normally distributed]]: <math display=inline>\sigma(X) \sim P( \mathcal{N}(\mu,\,\sigma^2) )</math>.
* The absolute value of {{tmath|X}} has [[folded normal distribution]]: <math display=inline>{{\left| X \right| \sim N_f(\mu, \sigma^2)}}</math>. If <math display=inline>\mu = 0</math> this is known as the [[half-normal distribution]].
* The absolute value of normalized residuals, <math display=inline>|X - \mu| / \sigma</math>, has [[chi distribution]] with one degree of freedom: <math display=inline>|X - \mu| / \sigma \sim \chi_1</math>.
* The square of <math display=inline>X/\sigma</math> has the [[noncentral chi-squared distribution]] with one degree of freedom: <math display=inline>X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)</math>. If <math display=inline>\mu = 0</math>, the distribution is called simply [[chi-squared distribution|chi-squared]].
* The log-likelihood of a normal variable {{tmath|x}} is simply the log of its [[probability density function]]: <math display=block>\ln p(x)= -\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2 -\ln \left(\sigma \sqrt{2\pi} \right).</math> Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted [[chi-squared distribution|chi-squared]] variable.
* The distribution of the variable {{tmath|X}} restricted to an interval <math display=inline>[a, b]</math> is called the [[truncated normal distribution]].
* <math display=inline>(X - \mu)^{-2}</math> has a [[Lévy distribution]] with location 0 and scale <math display=inline>\sigma^{-2}</math>.

===== Operations on two independent normal variables =====
* If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two [[independence (probability theory)|independent]] normal random variables, with means <math display=inline>\mu_1</math>, <math display=inline>\mu_2</math> and variances <math display=inline>\sigma_1^2</math>, <math display=inline>\sigma_2^2</math>, then their sum <math display=inline>X_1 + X_2</math> will also be normally distributed,<sup>[[sum of normally distributed random variables|[proof]]]</sup> with mean <math display=inline>\mu_1 + \mu_2</math> and variance <math display=inline>\sigma_1^2 + \sigma_2^2</math>.
* In particular, if {{tmath|X}} and {{tmath|Y}} are independent normal deviates with zero mean and variance <math display=inline>\sigma^2</math>, then <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent and normally distributed, with zero mean and variance <math display=inline>2\sigma^2</math>. This is a special case of the [[polarization identity]].<ref>{{harvtxt |Bryc |1995 |p=27 }}</ref>
* If <math display=inline>X_1</math>, <math display=inline>X_2</math> are two independent normal deviates with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, and {{tmath|a}}, {{tmath|b}} are arbitrary real numbers, then the variable <math display=block>
    X_3 = \frac{aX_1 + bX_2 - (a+b)\mu}{\sqrt{a^2+b^2}} + \mu
</math> is also normally distributed with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>. It follows that the normal distribution is [[stable distribution|stable]] (with exponent <math display=inline>\alpha=2</math>).
* If <math display=inline>X_k \sim \mathcal N(m_k, \sigma_k^2)</math>, <math display=inline>k \in \{ 0, 1 \}</math> are normal distributions, then their normalized [[geometric mean]] <math display=inline>\frac{1}{\int_{\R^n} X_0^{\alpha}(x) X_1^{1 - \alpha}(x) \, \text{d}x} X_0^{\alpha} X_1^{1 - \alpha}</math> is a normal distribution <math display=inline>\mathcal N(m_{\alpha}, \sigma_{\alpha}^2)</math> with <math display=inline>m_{\alpha} = \frac{\alpha m_0 \sigma_1^2 + (1 - \alpha) m_1 \sigma_0^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math> and <math display=inline>\sigma_{\alpha}^2 = \frac{\sigma_0^2 \sigma_1^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math>.

===== 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]].