Editing Normal distribution (section)

=== Operations and functions of normal variables ===

[[File:Probabilities of functions of normal vectors.png|thumb|right|'''a:''' Probability density of a function {{math|cos ''x''{{sup|2}}}} of a normal variable {{mvar|x}} with {{math|1= ''μ'' = −2}} and {{math|1= ''σ'' = 3}}. '''b:''' Probability density of a function {{mvar|x{{sup|y}}}} of two normal variables {{mvar|x}} and {{mvar|y}}, where {{math|1= ''μ{{sub|x}}'' = 1}}, {{math|1= ''μ{{sub|y}}'' = 2}}, {{math|1= ''σ{{sub|x}}'' = 0.1}}, {{math|1= ''σ{{sub|y}}'' = 0.2}}, and {{math|1= ''ρ{{sub|xy}}'' = 0.8}}. '''c:''' Heat map of the joint probability density of two functions of two correlated normal variables {{mvar|x}} and {{mvar|y}}, where {{math|1= ''μ{{sub|x}}'' = −2}}, {{math|1= ''μ{{sub|x}}'' = 5}}, {{math|1= {{subsup|σ|s=0|''x''|2}} = 10}}, {{math|1= {{subsup|σ|s=0|''y''|2}} = 20}}, and {{math|1= ''ρ{{sub|xy}}'' = 0.495}}. '''d:''' Probability density of a function {{math|{{abs|''x''{{sub|1}}}} + ... + {{abs|''x''{{sub|4}}}}}} of four [[iid]] standard normal variables. These are computed by the numerical method of ray-tracing.<ref name="Das-2021" />]]

The [[probability density]], [[cumulative distribution function|cumulative distribution]], and [[inverse cumulative distribution function|inverse cumulative distribution]] of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing<ref name="Das-2021">{{cite journal | last=Das|first=Abhranil| arxiv=2012.14331| title=A method to integrate and classify normal distributions|journal=Journal of Vision |date=2021|volume=21 |issue=10 |page=1 |doi=10.1167/jov.21.10.1 |pmid=34468706 |pmc=8419883 }}</ref> ([https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions Matlab code]). In the following sections we look at some special cases.

==== 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]].

==== Operations on multiple independent normal variables ====
* Any [[linear combination]] of independent normal deviates is a normal deviate.
* If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent standard normal random variables, then the sum of their squares has the [[chi-squared distribution]] with {{tmath|n}} degrees of freedom <math display=block>X_1^2 + \cdots + X_n^2 \sim \chi_n^2.</math>
* If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent normally distributed random variables with means {{tmath|\mu}} and variances <math display=inline>\sigma^2</math>, then their [[sample mean]] is independent from the sample [[standard deviation]],<ref>{{cite journal|title=A Characterization of the Normal Distribution |last=Lukacs |first=Eugene |journal=[[The Annals of Mathematical Statistics]] |issn=0003-4851 |volume=13|issue=1 |year=1942 |pages=91–3 |jstor=2236166 |doi=10.1214/aoms/1177731647 |doi-access=free}}</ref> which can be demonstrated using [[Basu's theorem]] or [[Cochran's theorem]].<ref>{{cite journal |title=On Some Characterizations of the Normal Distribution | last1=Basu|first1=D. |last2=Laha|first2=R. G.|journal=[[Sankhyā (journal)|Sankhyā]]|issn=0036-4452| volume=13|issue=4|year=1954|pages=359–62| jstor=25048183}}</ref> The ratio of these two quantities will have the [[Student's t-distribution]] with <math display=inline>n-1</math> degrees of freedom: <math display=block>t = \frac{\overline X - \mu}{S/\sqrt{n}} = \frac{\frac{1}{n}(X_1+\cdots+X_n) - \mu}{\sqrt{\frac{1}{n(n-1)}\left[(X_1-\overline X)^2 + \cdots+(X_n-\overline X)^2\right]}} \sim t_{n-1}.</math>
* If <math display=inline>X_1, X_2, \ldots, X_n</math>, <math display=inline>Y_1, Y_2, \ldots, Y_m</math> are independent standard normal random variables, then the ratio of their normalized sums of squares will have the [[F-distribution]] with {{math|(''n'', ''m'')}} degrees of freedom:<ref>{{cite book |title=Testing Statistical Hypotheses |edition=2nd | first=E. L. | last=Lehmann | publisher=Springer |year=1997 | isbn=978-0-387-94919-2| page=199}}</ref> <math display=block>F = \frac{\left(X_1^2+X_2^2+\cdots+X_n^2\right)/n}{\left(Y_1^2+Y_2^2+\cdots+Y_m^2\right)/m} \sim F_{n,m}.</math>

==== Operations on multiple correlated normal variables ====
* A [[quadratic form]] of a normal vector, i.e. a quadratic function <math display=inline>q = \sum x_i^2 + \sum x_j + c</math> of multiple independent or correlated normal variables, is a [[generalized chi-square distribution|generalized chi-square]] variable.