Editing Normal distribution (section)

=== Extensions ===
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called ''normal'' or ''Gaussian'' laws, so a certain ambiguity in names exists.
* The [[multivariate normal distribution]] describes the Gaussian law in the {{mvar|k}}-dimensional [[Euclidean space]]. A vector {{math|''X'' ∈ '''R'''<sup>''k''</sup>}} is multivariate-normally distributed if any linear combination of its components {{math|Σ{{su|p=''k''|b=''j''=1}}''a<sub>j</sub> X<sub>j</sub>''}} has a (univariate) normal distribution. The variance of {{mvar|X}} is a {{math|{{thinsp|''k''|×|''k''}}}} symmetric positive-definite matrix {{mvar|V}}. The multivariate normal distribution is a special case of the [[elliptical distribution]]s. As such, its iso-density loci in the {{math|1=''k'' = 2}} case are [[ellipse]]s and in the case of arbitrary {{mvar|k}} are [[ellipsoid]]s.
* [[Rectified Gaussian distribution]] a rectified version of normal distribution with all the negative elements reset to 0.
* [[Complex normal distribution]] deals with the complex normal vectors. A complex vector {{math|''X'' ∈ '''C'''<sup>''k''</sup>}} is said to be normal if both its real and imaginary components jointly possess a {{math|2''k''}}-dimensional multivariate normal distribution. The variance-covariance structure of {{mvar|X}} is described by two matrices: the ''{{dfn|variance}}'' matrix {{math|Γ}}, and the ''{{dfn|relation}}'' matrix {{mvar|C}}.
* [[Matrix normal distribution]] describes the case of normally distributed matrices.
* [[Gaussian process]]es are the normally distributed [[stochastic process]]es. These can be viewed as elements of some infinite-dimensional [[Hilbert space]] {{mvar|H}}, and thus are the analogues of multivariate normal vectors for the case {{math|''k'' {{=}} ∞}}. A random element {{math|''h'' ∈ ''H''}} is said to be normal if for any constant {{math|''a'' ∈ ''H''}} the [[scalar product]] {{math|(''a'', ''h'')}} has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear ''covariance operator'' {{math|''K'': ''H'' → ''H''}}. Several Gaussian processes became popular enough to have their own names:
** [[Wiener process|Brownian motion]];
** [[Brownian bridge]]; and
** [[Ornstein–Uhlenbeck process]].
* [[Gaussian q-distribution]] is an abstract mathematical construction that represents a [[q-analogue]] of the normal distribution.
* the [[q-Gaussian]] is an analogue of the Gaussian distribution, in the sense that it maximises the [[Tsallis entropy]], and is one type of [[Tsallis distribution]]. This distribution is different from the [[Gaussian q-distribution]] above.
* The [[Kaniadakis Gaussian distribution|Kaniadakis {{mvar|κ}}-Gaussian distribution]] is a generalization of the Gaussian distribution which arises from the [[Kaniadakis statistics]], being one of the [[Kaniadakis distribution]]s.

A random variable {{mvar|X}} has a two-piece normal distribution if it has a distribution

<math display=block>f_X( x ) = \begin{cases}
N( \mu, \sigma_1^2 ),& \text{ if } x \le \mu \\
N( \mu, \sigma_2^2 ),& \text{ if } x \ge \mu
\end{cases}</math>

where {{mvar|μ}} is the mean and {{math|{{subsup|''σ''|s=0|1|2}}}} and {{math|{{subsup|''σ''|s=0|2|2}}}} are the variances of the distribution to the left and right of the mean respectively.

The mean {{math|E(''X'')}}, variance {{math|V(''X'')}}, and third central moment {{math|T(''X'')}} of this distribution have been determined<ref name="John-1982">{{cite journal|last1=John|first1=S|year=1982|title=The three parameter two-piece normal family of distributions and its fitting|journal=Communications in Statistics – Theory and Methods|volume=11|issue=8|pages=879–885|doi=10.1080/03610928208828279}}</ref>

<math display=block>\begin{align}
\operatorname{E}( X ) &= \mu +  \sqrt{\frac 2 \pi } ( \sigma_2 - \sigma_1 ), \\
\operatorname{V}( X ) &= \left( 1 - \frac 2 \pi\right)( \sigma_2 - \sigma_1 )^2 + \sigma_1 \sigma_2, \\
\operatorname{T}( X ) &= \sqrt{ \frac 2 \pi}( \sigma_2 - \sigma_1 ) \left[ \left( \frac 4 \pi  - 1 \right) ( \sigma_2 - \sigma_1)^2 + \sigma_1 \sigma_2 \right].
\end{align}</math>

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
* [[Pearson distribution]] — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
* The [[generalized normal distribution]], also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.