en>Barmak.honarvar: /* Other representations */

2014-11-19T03:48:57Z

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128.101.152.66: /* Definitions */

2014-02-28T22:55:58Z

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146.155.21.168: /* Noll's sequential indices */ readability

2013-10-21T20:40:59Z

Noll's sequential indices: readability

← Older revision		Revision as of 22:40, 21 October 2013
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	~~Alyson Meagher is~~ the ~~title her parents gave her but she doesn~~'t ~~like when people use her complete name~~. ~~Mississippi~~ is the ~~only place I~~'~~ve been residing in but I will have~~ to ~~move in~~ a ~~year~~ or ~~two~~. He is ~~an order clerk~~ and it~~'s some thing he truly enjoy~~. ~~To perform lacross~~ is the ~~factor I adore most~~ of ~~all~~.<br><br>~~Also visit my web blog~~; [http://~~yoparo~~.org/~~content~~/~~hobby~~-~~one~~-~~thing~~-~~everyone~~-~~needs-have love psychics~~]		{{Probability distribution \|
			name =hyperbolic secant\|
			type =density\|
			pdf_image =[[Image:Sech_distribution_PDF.png\|325px\|Plot of the hyperbolic secant PDF]]\|
			cdf_image =[[Image:Sech_distribution_CDF.png\|325px\|Plot of the hyperbolic secant CDF]]\|
			parameters =''none''\|
			support =<math>x \in (-\infty; +\infty)\!</math>\|
			pdf =<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>\|
			cdf =<math>\frac{2}{\pi} \arctan\!\left[\exp\!\left(\frac{\pi}{2}\,x\right)\right]\!</math>\|
			mean =<math>0</math>\|
			median =<math>0</math>\|
			mode =<math>0</math>\|
			variance =<math>1</math>\|
			skewness =<math>0</math>\|
			kurtosis =<math>2</math>\|
			entropy =4/''π'' ''[[Catalan's constant\|K]]'' <math>\;\approx 1.16624</math>\|
			mgf =<math>\sec(t)\!</math> for <math>\|t\|<\frac{\pi}2\!</math>\|
			char =<math>\operatorname{sech}(t)\!</math> for <math>\|t\|<\frac{\pi}2\!</math>\|
			}}
			In [[probability theory]] and [[statistics]], the '''hyperbolic secant distribution''' is a continuous [[probability distribution]] whose [[probability density function]] and [[Characteristic function (probability theory)\|characteristic function]] are proportional to the [[hyperbolic function\|hyperbolic secant function]]. The hyperbolic secant function is equivalent to the inverse [[Hyperbolic function\|hyperbolic cosine]], and thus this distribution is also called the '''inverse-cosh distribution'''.

			== Explanation ==

			A [[random variable]] follows a hyperbolic secant distribution if its probability density function (pdf) can be related to the following standard form of density function by a location and shift transformation:

			:<math>f(x) = \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\! ,</math>

			where "sech" denotes the hyperbolic secant function.
			The [[cumulative distribution function]] (cdf) of the standard distribution is

			:<math>F(x) = \frac12 + \frac{1}{\pi} \arctan\!\left[\operatorname{sinh}\!\left(\frac{\pi}{2}\,x\right)\right]
			\! ,</math>
			:<math> = \frac{2}{\pi} \arctan\!\left[\exp\left(\frac{\pi}{2}\,x\right)\right] \! .</math>

			where "arctan" is the [[trigonometric function#Inverse functions\|inverse (circular) tangent function]].
			The inverse cdf (or quantile function) is

			:<math>F^{-1}(p) = -\frac{2}{\pi}\, \operatorname{arsinh}\!\left[\cot(\pi\,p)\right] \! ,</math>
			:<math> = \frac{2}{\pi}\, \ln\!\left[\tan\left(\frac{\pi}{2}\,p\right)\right] \! .</math>

			where "arsinh" is the [[inverse hyperbolic function\|inverse hyperbolic sine function]] and "cot" is the [[trigonometric function\|(circular) cotangent function]].

			The hyperbolic secant distribution shares many properties with the standard [[normal distribution]]: it is symmetric with unit [[variance]] and zero [[expected value\|mean]], [[median]] and [[mode (statistics)\|mode]], and its pdf is proportional to its characteristic function. However, the hyperbolic secant distribution is [[kurtosis#Terminology and examples\|leptokurtic]]; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution.

			Johnson et al. (1995, p147) place this distribution in the context of a class of generalised forms of the [[logistic distribution]], but use a different parameterisation of the standard distribution compared to that here.

			==References==
			* {{cite journal\|first=W. D. \|last=Baten \|year=1934 \|title=The probability law for the sum of ''n'' independent variables, each subject to the law <math>(2h)^{-1} \operatorname{sech}(\pi x/2h)</math> \|journal=[[Bulletin of the American Mathematical Society]] \|volume=40 \|issue=4 \|pages=284–290 \|doi=10.1090/S0002-9904-1934-05852-X }}
			* J. Talacko, 1956, "Perks' distributions and their role in the theory of Wiener's stochastic variables", ''[[Trabajos de Estadistica]]'' 7:159–174.
			* Luc Devroye, 1986, [http://cgm.cs.mcgill.ca/~luc/rnbookindex.html ''Non-Uniform Random Variate Generation''], Springer-Verlag, New York. Section IX.7.2.
			* {{Cite journal
			\| author = [[G.K. Smyth]]
			\| title = A note on modelling cross correlations: Hyperbolic secant regression
			\| journal = [[Biometrika]]
			\| volume = 81
			\| issue = 2
			\| pages = 396–402
			\| year = 1994
			\| url = http://www.statsci.org/smyth/pubs/sech.pdf
			\| doi = 10.1093/biomet/81.2.396
			}}
			* Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, ''Continuous Univariate Distributions'', volume 2, ISBN 0-471-58494-0.

			{{ProbDistributions\|continuous-infinite}}

			[[Category:Continuous distributions]]
			[[Category:Probability distributions]]

en>EleferenBot: r2.7.3) (Robot: Adding ru:Полиномы Цернике

2012-08-26T15:00:21Z

r2.7.3) (Robot: Adding ru:Полиномы Цернике

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Alyson Meagher is the title her parents gave her but she doesn't like when people use her complete name. Mississippi is the only place I've been residing in but I will have to move in a year or two. He is an order clerk and it's some thing he truly enjoy. To perform lacross is the factor I adore most of all.<br><br>Also visit my web blog; [http://yoparo.org/content/hobby-one-thing-everyone-needs-have love psychics]

Zernike polynomials - Revision history

en>Barmak.honarvar: /* Other representations */

128.101.152.66: /* Definitions */

146.155.21.168: /* Noll's sequential indices */ readability

en>EleferenBot: r2.7.3) (Robot: Adding ru:Полиномы Цернике