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| | | Some people choose to like their endeavor. Others learn to loathe gonna be work. Some people have the mentality it can be what what you are doing and whether you enjoy it or not is inconsequential. By choosing a wholesome pair of shoes or boots, you can build the day much more challenging.<br><br>First, absolutely set up car dvd player to listen to musical technology. Music is an effective companion uncover are operating a car. the car dvd player supports many mainstream formats such as: MP4, DVD, SVCD, CD, MP3, etc. Should are bored with uploding the music, you may use the high sensitive FM/AM tuner, FM/AM auto search, frequency render. With this function, you can catch at the top of the ugg news.<br><br>Later Uncovered that undoubtedly one of my colleagues often wore ugg boots, which are fashionable and light-weight. Even if we wore them typically the office, it wouldn't make us lose facial area. The best thing was that UGG boots were made from the pure wool, so wearing them were very warm and comfy. Besides if I couldn't want to use them within the office, I change then into our high heel shoes. So forth the way to work and home; suggest you always be very warm and trendy. In the cold winter, nothing will likely be more wonderful than turn out to be warm and trendy at once. ugg boots attracted my observation.<br><br>The beginning which usually anyone search out for to get these cheap boots is through the web. There are many shoe stores which give this footwear at affordable rates given that they seeking to help have more customers. In addition there are an assortment of online stores to choose from so occurrences compare coming from a number of stores and settle to get the best option one and the engineered so offers its boots at many affordable risk.<br><br>The UGG Classic Tall boots looks at style that goes to a max of the leg. It is made of Australian merino more attractive. To more meet women' s requirements, designers add colors like black, chestnut, chocolate, grey, sand, pink, romantic flower(the color loved by most girls), metallic gold and metallic pewter to them.<br><br>If you plan to order from factory directly, make sure the company is authorized. Some manufacturers provide you with sheepskin boots on incredibly cheap prices as those shoes are fakes. Therefore do research on official UGG websites before ordering.<br><br>If you loved this short article and you would such as to get even more info relating to [http://horizonafrica.com/img/ Cheap UGG Boots Australia Factory outlet online free shipping] kindly see the page. |
| The '''generalized normal distribution''' or '''generalized Gaussian distribution''' (GGD) is either of two families of [[parametric statistics|parametric]] [[continuous probability distribution]]s on the [[real number|real]] line. Both families add a [[shape parameter]] to the [[normal distribution]]. To distinguish the two families, they are referred to below as "version 1" and "version 2." However this is not a standard nomenclature.
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| ==Version 1==
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| {{Probability distribution |
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| name =Generalized Normal (version 1)|
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| type =density|
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| pdf_image =[[File:Generalized normal densities.svg|325px|Probability density plots of generalized normal distributions]]|
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| cdf_image =[[File:Generalized normal cdfs.svg|325px|Cumulative distribution function plots of generalized normal distributions]]|
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| parameters =<math> \mu \,</math> [[location parameter|location]] ([[real number|real]])<br/><math> \alpha \,</math> [[scale parameter|scale]] (positive, [[real number|real]])<br/><math> \beta \,</math> [[shape parameter|shape]] (positive, [[real number|real]])|
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| support =<math>x \in (-\infty; +\infty)\!</math>|
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| pdf =<math>\frac{\beta}{2\alpha\Gamma(1/\beta)} \; e^{-(|x-\mu|/\alpha)^\beta}</math> <br/><br/><math>\Gamma</math> denotes the [[gamma function]]|
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| cdf =<math>\frac{1}{2} + \sgn(x-\mu)\frac{\gamma\left[1/\beta, \left( \frac{|x-\mu|}{\alpha} \right)^\beta\right]}{2\Gamma(1/\beta)} </math> <br/>
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| <math>\gamma</math> denotes the lower [[incomplete gamma function]]|
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| mean =<math> \mu \,</math>|
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| median =<math> \mu \,</math>|
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| mode =<math> \mu \,</math>|
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| variance =<math>\frac{\alpha^2\Gamma(3/\beta)}{\Gamma(1/\beta)}</math>|
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| skewness =0|
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| kurtosis =<math>\frac{\Gamma(5/\beta)\Gamma(1/\beta)}{\Gamma(3/\beta)^2}-3</math>|
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| entropy =<math>\frac{1}{\beta}-\log\left[\frac{\beta}{2\alpha\Gamma(1/\beta)}\right]</math><ref>{{cite journal |last= Nadarajah|first= Saralees|authorlink= |coauthors= |date=September 2005|title= A generalized normal distribution|journal= Journal of Applied Statistics|volume= 32 |issue= 7|pages= 685–694|doi= 10.1080/02664760500079464 |url= |quote= }}</ref>|
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| mgf =<!-- to do -->|
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| char =<!-- to do -->|
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| }}
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| Known also as the '''exponential power distribution''', or the '''generalized error distribution''', this is a parametric family of symmetric distributions. It includes all [[normal distribution|normal]] and [[Laplace distribution|Laplace]] distributions, and as limiting cases it includes all [[continuous uniform distribution]]s on bounded intervals of the real line.
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| This family includes the [[normal distribution]] when <math>\textstyle\beta=2</math> (with mean <math>\textstyle\mu</math> and variance <math>\textstyle \frac{\alpha^2}{2}</math>) and it includes the [[Laplace distribution]] when <math>\textstyle\beta=1</math>. As <math>\textstyle\beta\rightarrow\infty</math>, the density [[pointwise convergence|converges pointwise]] to a uniform density on <math>\textstyle (\mu-\alpha,\mu+\alpha)</math>.
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| This family allows for tails that are either heavier than normal (when <math>\beta<2</math>) or lighter than normal (when <math>\beta>2</math>). It is a useful way to parametrize a continuum of symmetric, [[Platykurtic#Terminology and examples|platykurtic]] densities spanning from the normal (<math>\textstyle\beta=2</math>) to the uniform density (<math>\textstyle\beta=\infty</math>), and a continuum of symmetric, [[Platykurtic#Terminology and examples|leptokurtic]] densities spanning from the Laplace (<math>\textstyle\beta=1</math>) to the normal density (<math>\textstyle\beta=2</math>).
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| ===Parameter estimation===
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| Parameter estimation via [[maximum likelihood estimation|maximum likelihood]] and the [[method of moments]] has been studied.<ref>{{cite journal |last= Varanasi |first= M.K. |authorlink= |coauthors= Aazhang, B. |date=October 1989|title= Parametric generalized Gaussian density estimation|journal= Journal of the Acoustical Society of America|volume= 86|issue= 4|pages= 1404–1415|id= |url= |doi= 10.1121/1.398700}}</ref> The estimates do not have a closed form and must be obtained numerically. Estimators that do not require numerical calculation have also been proposed.<ref>
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| {{cite journal |last= Domínguez-Molina |first= J. Armando|authorlink= |coauthors= González-Farías, Graciela; Rodríguez-Dagnino, Ramón M. | title= A practical procedure to estimate the shape parameter in the generalized Gaussian distribution | url= http://www.cimat.mx/reportes/enlinea/I-01-18_eng.pdf |accessdate=2009-03-03 }}</ref>
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| The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C<sup>∞</sup> of [[smooth function]]s) only if <math>\textstyle\beta</math> is a positive, even integer. Otherwise, the function has <math>\textstyle\lfloor \beta \rfloor</math> continuous derivatives. As a result, the standard results for consistency and asymptotic normality of [[maximum likelihood]] estimates of <math>\beta</math> only apply when <math>\textstyle\beta\ge 2</math>.
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| ==== Maximum likelihood estimator ====
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| It is possible to fit the generalized normal distribution adopting an approximate [[maximum likelihood]] method.<ref>{{cite journal |last= Varanasi|first= M.K.|coauthors= Aazhang B. |year= 1989|month= |title= Parametric generalized Gaussian density estimation|journal= [[J. Acoust. Soc. Am.]] |volume= 86|issue= |pages= 1404–1415|id= |url= |doi= }}</ref><ref>{{cite journal |last= Do |first= M.N.|coauthors= Vetterli, M. |date=February 2002|title= Wavelet-based Texture Retrival Using Generalised Gaussian Density and Kullback-Leibler Distance|journal= Transaction on Image Processing|volume= 11|issue= |pages= 146–158|id= |url= |doi= }}</ref> With <math>\mu</math> initially set to the sample first moment <math>m_1</math>,
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| <math>\textstyle\beta</math> is estimated by using a [[Newton's method|Newton-Raphson]] iterative procedure, starting from an initial guess of <math>\textstyle\beta=\textstyle\beta_0</math>,
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| :<math>\beta _0 = \frac{m_1}{\sqrt{m_2}},</math>
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| where
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| :<math>m_1={1 \over N} \sum_{i=1}^N |x_i|,</math>
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| is the first statistical [[Moment (mathematics)|moment]] of the absolute values and <math>m_2</math> is the second statistical [[Moment (mathematics)|moment]]. The iteration is
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| :<math>\beta _{i+1} = \beta _{i} - \frac{g(\beta _{i})}{g'(\beta _{i})} ,</math>
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| where
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| :<math>g(\beta)= 1 + \frac{\psi(1/\beta)}{\beta} - \frac{\sum_{i=1}^{N} |x_i-\mu|^{\beta} \log|x_i-\mu| }{\sum_{i=1}^{N} |x_i-\mu|^{\beta}} + \frac{\log( \frac{\beta}{N} \sum_{i=1}^{N} |x_i-\mu|^{\beta})}{\beta} ,</math>
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| :<math>g'(\beta)= -\frac{\psi(1/\beta)}{\beta^2} - \frac{\psi'(1/\beta)}{\beta^3} + \frac{1}{\beta^2} - \frac{\sum_{i=1}^{N} |x_i-\mu|^{\beta} (\log|x_i-\mu|)^2}{\sum_{i=1}^{N} |x_i-\mu|^{\beta}} + \frac{(\sum_{i=1}^{N} |x_i-\mu|^{\beta} \log|x_i-\mu|)^2}{(\sum_{i=1}^{N} |x_i-\mu|^{\beta})^2} + \frac{\sum_{i=1}^{N} |x_i-\mu|^{\beta} \log|x_i-\mu|}{\beta \sum_{i=1}^{N} |x_i-\mu|^{\beta}} - \frac{\log(\frac{\beta}{N} \sum_{i=1}^{N} |x_i-\mu|^{\beta} )}{\beta^2} ,</math>
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| and where <math>\psi()</math> and <math>\psi'()</math> are the [[digamma function]] and [[trigamma function]]. | |
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| Given a value for <math>\textstyle\beta</math>, it is possible to estimate <math>\mu</math> by finding the minimum of:
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| :<math>min_{\mu}=\sum_{i=1}^{N} |x_i-\mu|^{\beta}</math>
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| Finally <math>\textstyle\alpha</math> is evaluated as
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| :<math>\alpha = ( \frac{\beta}{N} \sum_{i=1}^{N}|x_i-\mu|^{\beta})^{\frac{1}{ \beta}} .</math>
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| ===Applications===
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| This version of the generalized normal distribution has been used in modeling when the concentration of values around the mean and the tail behavior are of particular interest.<ref>
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| {{cite journal |last= Liang|first= Faming|authorlink= |coauthors= Liu, Chuanhai; Wang, Naisyin
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| |date=April 2007|title= A robust sequential Bayesian method for identification of differentially expressed genes|journal= Statistica Sinica|volume= 17|issue= 2|pages= 571–597|id= |url= http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=17&num=2&art=8 |accessdate=2009-03-03 |quote= }}</ref><ref>
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| {{cite book |title= Bayesian Inference in Statistical Analysis |last= Box |first= George E. P.|authorlink= George E. P. Box |coauthors= Tiao, George C. |year= 1992 |publisher= Wiley|location= New York|isbn= 0-471-57428-7|page= |pages= |url= }}</ref> Other families of distributions can be used if the focus is on other deviations from normality. If the [[Symmetric distribution|symmetry]] of the distribution is the main interest, the [[skew normal distribution|skew normal]] family or version 2 of the generalized normal family discussed below can be used. If the tail behavior is the main interest, the [[student t distribution|student t]] family can be used, which approximates the normal distribution as the degrees of freedom grows to infinity. The t distribution, unlike this generalized normal distribution, obtains heavier than normal tails without acquiring a [[cusp (singularity)|cusp]] at the origin.
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| ===Properties===
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| The multivariate generalized normal distribution, i.e. the product of <math>n</math> exponential power distributions with the same <math>\beta</math> and <math>\alpha</math> parameters, is the only probability density that can be written in the form <math>p(\mathbf x)=g(\|\mathbf x\|_\beta)</math> and has independent marginals.<ref>{{cite journal |last= Sinz|first= Fabian|authorlink= |coauthors= Gerwinn, Sebastian; Bethge, Matthias
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| |date=May 2009|title=Characterization of the p-Generalized Normal Distribution. |journal=Journal of Multivariate Analysis|volume= 100|issue= 5|pages= 817–820|id= |accessdate= |quote= |doi=10.1016/j.jmva.2008.07.006}}</ref> The results for the special case of the [[Multivariate normal distribution]] is originally attributed to [[James Clerk Maxwell|Maxwell]].<ref>{{cite journal |last= Kac|first= M.|authorlink= |coauthors=|year= 1939|month= |title=On a characterization of the normal distribution|journal=American Journal of Mathematics|volume= 61|issue= 3|pages= 726–728|id= |url= |accessdate= |quote= |doi= 10.2307/2371328 }}</ref>
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| ==Version 2==
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| {{Probability distribution |
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| name =Generalized Normal (version 2)|
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| type =density|
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| pdf_image =[[File:Generalized normal densities 2.svg|325px|Probability density plots of generalized normal distributions]]|
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| cdf_image =[[File:Generalized normal cdfs 2.svg|325px|Cumulative distribution function plots of generalized normal distributions]]|
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| parameters =<math> \xi \,</math> [[location parameter|location]] ([[real number|real]])<br/><math> \alpha \,</math> [[scale parameter|scale]] (positive, [[real number|real]])<br/><math> \kappa \,</math> [[shape parameter|shape]] ([[real number|real]])|
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| support =<math>x \in (-\infty,\xi+\alpha/\kappa) \text{ if } \kappa>0</math><br/><math>x \in (-\infty,\infty) \text{ if } \kappa=0</math><br/><math>x \in (\xi+\alpha/\kappa; +\infty) \text{ if } \kappa<0</math>|
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| pdf =<math> \frac{\phi(y)}{\alpha-\kappa(x-\xi)}</math>, where <br/><math>y = \begin{cases} - \frac{1}{\kappa} \log \left[ 1- \frac{\kappa(x-\xi)}{\alpha} \right] & \text{if } \kappa \neq 0 \\ \frac{x-\xi}{\alpha} & \text{if } \kappa=0 \end{cases} </math><br><math>\phi</math> is the standard [[normal distribution|normal]] [[probability distribution function|pdf]]|
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| cdf =<math> \Phi(y) </math>, where <br/><math> y = \begin{cases} - \frac{1}{\kappa} \log \left[ 1- \frac{\kappa(x-\xi)}{\alpha} \right] & \text{if } \kappa \neq 0 \\ \frac{x-\xi}{\alpha} & \text{if } \kappa=0 \end{cases} </math><br><math>\Phi</math> is the standard [[normal distribution|normal]] [[cumulative distribution function|CDF]]|
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| mean =<math>\xi - \frac{\alpha}{\kappa} \left( e^{\kappa^2/2} - 1 \right)</math>|
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| median =<math>\xi \,</math>|
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| mode =<!-- to do -->|
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| variance =<math>\frac{\alpha^2}{\kappa^2} e^{\kappa^2} \left( e^{\kappa^2} - 1 \right)</math>|
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| skewness =<math>\frac{3 e^{\kappa^2} - e^{3 \kappa^2} - 2}{(e^{\kappa^2} - 1)^{3/2}} \text{ sign}(\kappa) </math>|
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| kurtosis =<math>e^{4 \kappa^2} + 2 e^{3 \kappa^2} + 3 e^{2 \kappa^2} - 6 </math>|
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| entropy =<!-- to do -->|
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| mgf =<!-- to do -->|
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| char =<!-- to do -->|
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| }}
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| This is a family of continuous probability distributions in which the shape parameter can be used to introduce skew.<ref>Hosking, J.R.M., Wallis, J.R. (1997) ''Regional frequency analysis: an approach based on L-moments'', Cambridge University Press. ISBN 0-521-43045-3. Section A.8</ref><ref>[http://bm2.genes.nig.ac.jp/RGM2/R_current/library/lmomco/man/cdfgno.html Documentation for the lmomco R package]</ref> When the shape parameter is zero, the normal distribution results. Positive values of the shape parameter yield left-skewed distributions bounded to the right, and negative values of the shape parameter yield right-skewed distributions bounded to the left. Only when the shape parameter is zero is the density function for this distribution positive over the whole real line: in this case the distribution is a [[normal distribution]], otherwise the distributions are shifted and possibly reversed [[log-normal distribution]]s.
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| ===Parameter estimation===
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| Parameters can be estimated via [[maximum likelihood estimation]] or the method of moments. The parameter estimates do not have a closed form, so numerical calculations must be used to compute the estimates. Since the sample space (the set of real numbers where the density is non-zero) depends on the true value of the parameter, some standard results about the performance of parameter estimates will not automatically apply when working with this family.
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| ===Applications===
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| This family of distributions can be used to model values that may be normally distributed, or that may be either right-skewed or left-skewed relative to the normal distribution. The [[skew normal distribution]] is another distribution that is useful for modeling deviations from normality due to skew. Other distributions used to model skewed data include the [[gamma distribution|gamma]], [[lognormal distribution|lognormal]], and [[Weibull distribution|Weibull]] distributions, but these do not include the normal distributions as special cases.
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| ==Other distributions related to the normal==
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| The two generalized normal families described here, like the [[skew normal distribution|skew normal]] family, are parametric families that extends the normal distribution by adding a shape parameter. Due to the central role of the normal distribution in probability and statistics, many distributions can be characterized in terms of their relationship to the normal distribution. For example, the [[lognormal distribution|lognormal]], [[folded normal distribution|folded normal]], and [[inverse normal distribution|inverse normal]] distributions are defined as transformations of a normally-distributed value, but unlike the generalized normal and skew-normal families, these do not include the normal distributions as special cases.
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| ==See also==
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| * [[Skew normal distribution]]
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| ==References==
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| {{reflist}}
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| {{ProbDistributions|continuous}}
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| {{Statistics|hide}}
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| {{DEFAULTSORT:Generalized Normal Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Normal distribution| ]]
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| [[Category:Probability distributions]]
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