Editing Normal distribution (section)

=== Other properties ===
{{ordered list
| 1 = If the characteristic function <math display=inline>\phi_X</math> of some random variable {{tmath|X}} is of the form <math display=inline>\phi_X(t) = \exp Q(t)</math> in a neighborhood of zero, where <math display=inline>Q(t)</math> is a [[polynomial]], then the '''Marcinkiewicz theorem''' (named after [[Józef Marcinkiewicz]]) asserts that {{tmath|Q}} can be at most a quadratic polynomial, and therefore {{tmath|X}} is a normal random variable.<ref name="Bryc 1995 35" /> The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero [[cumulant]]s.
| 2 = If {{tmath|X}} and {{tmath|Y}} are [[jointly normal]] and [[uncorrelated]], then they are [[independence (probability theory)|independent]]. The requirement that {{tmath|X}} and {{tmath|Y}} should be ''jointly'' normal is essential; without it the property does not hold.<ref>[http://www.math.uiuc.edu/~r-ash/Stat/StatLec21-25.pdf UIUC, Lecture 21. ''The Multivariate Normal Distribution''], 21.6:"Individually Gaussian Versus Jointly Gaussian".</ref><ref>Edward L. Melnick and Aaron Tenenbein, "Misspecifications of the Normal Distribution", ''[[The American Statistician]]'', volume 36, number 4 November 1982, pages 372–373</ref><sup>[[Normally distributed and uncorrelated does not imply independent|[proof]]]</sup> For non-normal random variables uncorrelatedness does not imply independence.
| 3 = The [[Kullback–Leibler divergence]] of one normal distribution <math display=inline>X_1 \sim N(\mu_1, \sigma^2_1)</math> from another <math display=inline>X_2 \sim N(\mu_2, \sigma^2_2)</math> is given by:<ref>{{cite web |url=http://www.allisons.org/ll/MML/KL/Normal/|title=Kullback Leibler (KL) Distance of Two Normal (Gaussian) Probability Distributions| website=Allisons.org |date=2007-12-05 |access-date=2017-03-03}}</ref> <math display=block>
    D_\mathrm{KL}( X_1 \parallel X_2 ) = \frac{(\mu_1 - \mu_2)^2}{2\sigma_2^2} + \frac{1}{2}\left( \frac{\sigma_1^2}{\sigma_2^2} - 1 - \ln\frac{\sigma_1^2}{\sigma_2^2} \right)
</math>
The [[Hellinger distance]] between the same distributions is equal to <math display=block>
    H^2(X_1,X_2) = 1 - \sqrt{\frac{2\sigma_1\sigma_2}{\sigma_1^2+\sigma_2^2}} \exp\left(-\frac{1}{4}\frac{(\mu_1-\mu_2)^2}{\sigma_1^2+\sigma_2^2}\right)
</math>
| 4 = The [[Fisher information matrix]] for a normal distribution w.r.t. {{tmath|\mu}} and <math display=inline>\sigma^2</math> is diagonal and takes the form <math display=block>
    \mathcal I (\mu, \sigma^2) = \begin{pmatrix} \frac{1}{\sigma^2} & 0 \\ 0 & \frac{1}{2\sigma^4} \end{pmatrix}
</math>
| 5 = The [[conjugate prior]] of the mean of a normal distribution is another normal distribution.<ref>{{cite web|url=http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture5.pdf|title=Stat260: Bayesian Modeling and Inference: The Conjugate Prior for the Normal Distribution|first=Michael I.|last=Jordan|date=February 8, 2010}}</ref> Specifically, if <math display=inline>x_1, \ldots, x_n</math> are iid <math display=inline>\sim N(\mu, \sigma^2)</math> and the prior is <math display=inline>\mu \sim N(\mu_0 , \sigma^2_0)</math>, then the posterior distribution for the estimator of {{tmath|\mu}} will be <math display=block>
    \mu \mid x_1,\ldots,x_n \sim \mathcal{N}\left( \frac{\frac{\sigma^2}{n}\mu_0 + \sigma_0^2\bar{x}}{\frac{\sigma^2}{n}+\sigma_0^2},\left( \frac{n}{\sigma^2} + \frac{1}{\sigma_0^2} \right)^{-1} \right)
</math>
| 6 = The family of normal distributions not only forms an [[exponential family]] (EF), but in fact forms a [[natural exponential family]] (NEF) with quadratic [[variance function]] ([[NEF-QVF]]). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF.
| 7 = In [[information geometry]], the family of normal distributions forms a [[statistical manifold]] with [[constant curvature]] {{tmath|-1}}. The same family is [[flat manifold|flat]] with respect to the (±1)-connections <math display=inline>\nabla^{(e)}</math> and <math display=inline>\nabla^{(m)}</math>.<ref>{{harvtxt |Amari |Nagaoka |2000 }}</ref>
| 8 = If <math display=inline>X_1, \dots, X_n</math> are distributed according to <math display=inline>N(0, \sigma^2)</math>, then <math display=inline>E[\max_i X_i ] \leq \sigma\sqrt{2\ln n}</math>. Note that there is no assumption of independence.<ref>{{Cite web |title=Expectation of the maximum of gaussian random variables |url=https://math.stackexchange.com/a/89147 |access-date=2024-04-07 |website=Mathematics Stack Exchange |language=en}}</ref>
}}