Bessel polynomials - Revision history

en>OccultZone: fix CS1 error

2014-11-30T11:56:07Z

fix CS1 error

← Older revision		Revision as of 13:56, 30 November 2014
Line 1:		Line 1:
	~~{{Refimprove\|date=December 2009}}~~		Hi there. My name is Sophia Meagher although it is not the title on my beginning certification. Credit authorising is where my primary earnings comes from. I've always cherished residing in Mississippi. To play lacross is the factor I love most of all.<br><br>my site; [http://www.sirudang.com/siroo_Notice/2110 online psychic chat]
	~~{{Probability distribution \|~~
	name ~~=Normal-inverse Gaussian (NIG)\|~~
	~~type =density\|~~
	~~pdf_image =\|~~
	~~cdf_image =\|~~
	parameters =<math>\mu</math> [[location parameter\|location]] ([[real number\|real]])<br/> <math>\alpha</math> tail heavyness (real)<br/> <math>\beta</math> asymmetry parameter (real)<br/> <math>\delta</math> [[scale parameter]] (real)<br/> <math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>\|
	~~support =<math>x \in (-\infty; +\infty)\!</math>\|~~
	pdf =<math>\frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}</math> <br/><br/> <math>K_j</math> denotes a modified [[Bessel function]] of the second kind\|
	~~cdf =<!-- to do -->\|~~
	~~mean =<math>\mu + \delta \beta / \gamma</math>\|~~
	~~median =<!-- to do -->\|~~
	~~mode =\|~~
	~~variance =<math>\delta\alpha^2/\gamma^3</math>\|~~
	~~skewness =<math> 3 \beta /(\alpha \sqrt{\delta \gamma})</math>\|~~
	~~kurtosis =<math>3(1+4 \beta^2/\alpha^2)/(\delta\gamma)</math>\|~~
	~~entropy =<!-- to do -->\|~~
	~~mgf =<math>e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}</math>\|~~
	~~char =<math>e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}</math>\|~~
	}}

	~~The '''normal-inverse Gaussian distribution (NIG)'''~~ is ~~[[continuous probability distribution]] that is defined as the [[normal variance-mean mixture]] where the mixing density~~ is the ~~[[inverse Gaussian distribution]]. The NIG distribution was introduced by [[Ole Barndorff-Nielsen]]<ref>{{cite journal\|doi=10.1098/rspa.1977.0041\|last=Barndorff-Nielsen\|first=Ole\|year=1977\|~~title~~=Exponentially decreasing distributions for the logarithm of particle size\|journal=Proceedings of the Royal Society of London~~. ~~Series A, Mathematical and Physical Sciences\|volume=353\|issue=1674\|pages=401–409\|jstor=79167\|publisher=The Royal Society}}</ref> and~~ is ~~a subclass of the [[generalised hyperbolic distribution]]~~. The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the [[normal distribution]], <math>N(\mu,\sigma^2),</math> arises as a special case by setting <math>\beta=0, \delta=\sigma^2\alpha,</math> and letting <math>\alpha\rightarrow\infty</math>.

	~~The fact that there~~ is ~~a simple expression for~~ the ~~moment generating function implies that simple expressions for~~ all ~~moments are available~~. ~~The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If~~ <~~math~~>~~X_1~~<~~/math~~> ~~and <math>X_2</math> are~~ [[statistical independence\|independent]] [[random variable]]s that are NIG-distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the location and scale parameters, <math>\mu_1</math>, <math>\delta_1</math> and <math>\mu_2,</math> <math>\delta_2</math>, respectively, then <math>X_1 + X_2</math> is NIG-distributed with parameters <math>\alpha, </~~math> <math>\beta, <~~/~~math><math>\mu_1+\mu_2</math> and <math>\delta_1 + \delta_2~~.~~</math>~~

	The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion ([[Wiener process]]), <math>W^{(\gamma)}(t)=W(t)+\gamma t</math>, we can define the inverse Gaussian process <math>A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.</~~math> Then given a second independent drifting Brownian motion, <math>W^{(\beta)}(t)=\tilde W(t)+\beta t<~~/math>, the normal-inverse Gaussian process is the time-changed process <math>X_t=W^{(\beta)}(A_t)</math>. The process <math>X(t)</math> at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of [[Lévy processes]].

	~~The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.~~

	~~==References==~~
	~~<references/>~~

	~~{{ProbDistributions\|continuous-infinite}}~~

	~~{{DEFAULTSORT:Normal-Inverse Gaussian Distribution}}~~
	~~[[Category:Continuous distributions]]~~
	~~[[Category:Generalized hyperbolic distributions]]~~
	~~[[Category:Probability distributions]~~]

en>Slawekb: Revert. The article needs references, not more OR. Undid revision 536215919 by A. Pichler (talk)

2013-03-28T00:47:18Z

Revert. The article needs references, not more OR. Undid revision 536215919 by A. Pichler (talk)

← Older revision		Revision as of 02:47, 28 March 2013
Line 1:		Line 1:
	The ~~author~~'s title is ~~Christy Brookins~~. ~~Alaska~~ is ~~exactly where I've always been residing~~. ~~Office supervising~~ is ~~where my primary income arrives from~~ but ~~I've usually wanted my own company~~. ~~My husband doesn't like it~~ the way ~~I do but what I really like doing is caving but I don'~~t ~~have~~ the ~~time lately~~.<br><br>~~My blog; [http:~~//~~www~~.~~article~~-~~galaxy~~.~~com/profile~~.~~php?~~a=~~143251 certified psychics~~]		{{Refimprove\|date=December 2009}}
			{{Probability distribution \|
			name =Normal-inverse Gaussian (NIG)\|
			type =density\|
			pdf_image =\|
			cdf_image =\|
			parameters =<math>\mu</math> [[location parameter\|location]] ([[real number\|real]])<br/> <math>\alpha</math> tail heavyness (real)<br/> <math>\beta</math> asymmetry parameter (real)<br/> <math>\delta</math> [[scale parameter]] (real)<br/> <math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>\|
			support =<math>x \in (-\infty; +\infty)\!</math>\|
			pdf =<math>\frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}</math> <br/><br/> <math>K_j</math> denotes a modified [[Bessel function]] of the second kind\|
			cdf =<!-- to do -->\|
			mean =<math>\mu + \delta \beta / \gamma</math>\|
			median =<!-- to do -->\|
			mode =\|
			variance =<math>\delta\alpha^2/\gamma^3</math>\|
			skewness =<math> 3 \beta /(\alpha \sqrt{\delta \gamma})</math>\|
			kurtosis =<math>3(1+4 \beta^2/\alpha^2)/(\delta\gamma)</math>\|
			entropy =<!-- to do -->\|
			mgf =<math>e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}</math>\|
			char =<math>e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}</math>\|
			}}

			The '''normal-inverse Gaussian distribution (NIG)''' is [[continuous probability distribution]] that is defined as the [[normal variance-mean mixture]] where the mixing density is the [[inverse Gaussian distribution]]. The NIG distribution was introduced by [[Ole Barndorff-Nielsen]]<ref>{{cite journal\|doi=10.1098/rspa.1977.0041\|last=Barndorff-Nielsen\|first=Ole\|year=1977\|title=Exponentially decreasing distributions for the logarithm of particle size\|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences\|volume=353\|issue=1674\|pages=401–409\|jstor=79167\|publisher=The Royal Society}}</ref> and is a subclass of the [[generalised hyperbolic distribution]]. The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the [[normal distribution]], <math>N(\mu,\sigma^2),</math> arises as a special case by setting <math>\beta=0, \delta=\sigma^2\alpha,</math> and letting <math>\alpha\rightarrow\infty</math>.

			The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If <math>X_1</math> and <math>X_2</math> are [[statistical independence\|independent]] [[random variable]]s that are NIG-distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the location and scale parameters, <math>\mu_1</math>, <math>\delta_1</math> and <math>\mu_2,</math> <math>\delta_2</math>, respectively, then <math>X_1 + X_2</math> is NIG-distributed with parameters <math>\alpha, </math> <math>\beta, </math><math>\mu_1+\mu_2</math> and <math>\delta_1 + \delta_2.</math>

			The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion ([[Wiener process]]), <math>W^{(\gamma)}(t)=W(t)+\gamma t</math>, we can define the inverse Gaussian process <math>A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.</math> Then given a second independent drifting Brownian motion, <math>W^{(\beta)}(t)=\tilde W(t)+\beta t</math>, the normal-inverse Gaussian process is the time-changed process <math>X_t=W^{(\beta)}(A_t)</math>. The process <math>X(t)</math> at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of [[Lévy processes]].

			The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.

			==References==
			<references/>

			{{ProbDistributions\|continuous-infinite}}

			{{DEFAULTSORT:Normal-Inverse Gaussian Distribution}}
			[[Category:Continuous distributions]]
			[[Category:Generalized hyperbolic distributions]]
			[[Category:Probability distributions]]

en>Suslindisambiguator at 17:02, 20 August 2012

2012-08-20T17:02:43Z

New page

The author's title is Christy Brookins. Alaska is exactly where I've always been residing. Office supervising is where my primary income arrives from but I've usually wanted my own company. My husband doesn't like it the way I do but what I really like doing is caving but I don't have the time lately.<br><br>My blog; [http://www.article-galaxy.com/profile.php?a=143251 certified psychics]

← Older revision		Revision as of 13:56, 30 November 2014
Line 1:		Line 1:
	~~{{Refimprove\|date=December 2009}}~~		Hi there. My name is Sophia Meagher although it is not the title on my beginning certification. Credit authorising is where my primary earnings comes from. I've always cherished residing in Mississippi. To play lacross is the factor I love most of all.<br><br>my site; [http://www.sirudang.com/siroo_Notice/2110 online psychic chat]
	~~{{Probability distribution \|~~
	name ~~=Normal-inverse Gaussian (NIG)\|~~
	~~type =density\|~~
	~~pdf_image =\|~~
	~~cdf_image =\|~~
	parameters =<math>\mu</math> [[location parameter\|location]] ([[real number\|real]])<br/> <math>\alpha</math> tail heavyness (real)<br/> <math>\beta</math> asymmetry parameter (real)<br/> <math>\delta</math> [[scale parameter]] (real)<br/> <math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>\|
	~~support =<math>x \in (-\infty; +\infty)\!</math>\|~~
	pdf =<math>\frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}</math> <br/><br/> <math>K_j</math> denotes a modified [[Bessel function]] of the second kind\|
	~~cdf =<!-- to do -->\|~~
	~~mean =<math>\mu + \delta \beta / \gamma</math>\|~~
	~~median =<!-- to do -->\|~~
	~~mode =\|~~
	~~variance =<math>\delta\alpha^2/\gamma^3</math>\|~~
	~~skewness =<math> 3 \beta /(\alpha \sqrt{\delta \gamma})</math>\|~~
	~~kurtosis =<math>3(1+4 \beta^2/\alpha^2)/(\delta\gamma)</math>\|~~
	~~entropy =<!-- to do -->\|~~
	~~mgf =<math>e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}</math>\|~~
	~~char =<math>e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}</math>\|~~
	}}

	~~The '''normal-inverse Gaussian distribution (NIG)'''~~ is ~~[[continuous probability distribution]] that is defined as the [[normal variance-mean mixture]] where the mixing density~~ is the ~~[[inverse Gaussian distribution]]. The NIG distribution was introduced by [[Ole Barndorff-Nielsen]]<ref>{{cite journal\|doi=10.1098/rspa.1977.0041\|last=Barndorff-Nielsen\|first=Ole\|year=1977\|~~title~~=Exponentially decreasing distributions for the logarithm of particle size\|journal=Proceedings of the Royal Society of London~~. ~~Series A, Mathematical and Physical Sciences\|volume=353\|issue=1674\|pages=401–409\|jstor=79167\|publisher=The Royal Society}}</ref> and~~ is ~~a subclass of the [[generalised hyperbolic distribution]]~~. The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the [[normal distribution]], <math>N(\mu,\sigma^2),</math> arises as a special case by setting <math>\beta=0, \delta=\sigma^2\alpha,</math> and letting <math>\alpha\rightarrow\infty</math>.

	~~The fact that there~~ is ~~a simple expression for~~ the ~~moment generating function implies that simple expressions for~~ all ~~moments are available~~. ~~The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If~~ <~~math~~>~~X_1~~<~~/math~~> ~~and <math>X_2</math> are~~ [[statistical independence\|independent]] [[random variable]]s that are NIG-distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the location and scale parameters, <math>\mu_1</math>, <math>\delta_1</math> and <math>\mu_2,</math> <math>\delta_2</math>, respectively, then <math>X_1 + X_2</math> is NIG-distributed with parameters <math>\alpha, </~~math> <math>\beta, <~~/~~math><math>\mu_1+\mu_2</math> and <math>\delta_1 + \delta_2~~.~~</math>~~

	The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion ([[Wiener process]]), <math>W^{(\gamma)}(t)=W(t)+\gamma t</math>, we can define the inverse Gaussian process <math>A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.</~~math> Then given a second independent drifting Brownian motion, <math>W^{(\beta)}(t)=\tilde W(t)+\beta t<~~/math>, the normal-inverse Gaussian process is the time-changed process <math>X_t=W^{(\beta)}(A_t)</math>. The process <math>X(t)</math> at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of [[Lévy processes]].

	~~The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.~~

	~~==References==~~
	~~<references/>~~

	~~{{ProbDistributions\|continuous-infinite}}~~

	~~{{DEFAULTSORT:Normal-Inverse Gaussian Distribution}}~~
	~~[[Category:Continuous distributions]]~~
	~~[[Category:Generalized hyperbolic distributions]]~~
	~~[[Category:Probability distributions]~~]

← Older revision		Revision as of 02:47, 28 March 2013
Line 1:		Line 1:
	The ~~author~~'s title is ~~Christy Brookins~~. ~~Alaska~~ is ~~exactly where I've always been residing~~. ~~Office supervising~~ is ~~where my primary income arrives from~~ but ~~I've usually wanted my own company~~. ~~My husband doesn't like it~~ the way ~~I do but what I really like doing is caving but I don'~~t ~~have~~ the ~~time lately~~.<br><br>~~My blog; [http:~~//~~www~~.~~article~~-~~galaxy~~.~~com/profile~~.~~php?~~a=~~143251 certified psychics~~]		{{Refimprove\|date=December 2009}}
			{{Probability distribution \|
			name =Normal-inverse Gaussian (NIG)\|
			type =density\|
			pdf_image =\|
			cdf_image =\|
			parameters =<math>\mu</math> [[location parameter\|location]] ([[real number\|real]])<br/> <math>\alpha</math> tail heavyness (real)<br/> <math>\beta</math> asymmetry parameter (real)<br/> <math>\delta</math> [[scale parameter]] (real)<br/> <math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>\|
			support =<math>x \in (-\infty; +\infty)\!</math>\|
			pdf =<math>\frac{\alpha\delta K_1 \left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)}{\pi \sqrt{\delta^2 + (x - \mu)^2}} \; e^{\delta \gamma + \beta (x - \mu)}</math> <br/><br/> <math>K_j</math> denotes a modified [[Bessel function]] of the second kind\|
			cdf =<!-- to do -->\|
			mean =<math>\mu + \delta \beta / \gamma</math>\|
			median =<!-- to do -->\|
			mode =\|
			variance =<math>\delta\alpha^2/\gamma^3</math>\|
			skewness =<math> 3 \beta /(\alpha \sqrt{\delta \gamma})</math>\|
			kurtosis =<math>3(1+4 \beta^2/\alpha^2)/(\delta\gamma)</math>\|
			entropy =<!-- to do -->\|
			mgf =<math>e^{\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +z)^2})}</math>\|
			char =<math>e^{i\mu z + \delta (\gamma - \sqrt{\alpha^2 -(\beta +iz)^2})}</math>\|
			}}

			The '''normal-inverse Gaussian distribution (NIG)''' is [[continuous probability distribution]] that is defined as the [[normal variance-mean mixture]] where the mixing density is the [[inverse Gaussian distribution]]. The NIG distribution was introduced by [[Ole Barndorff-Nielsen]]<ref>{{cite journal\|doi=10.1098/rspa.1977.0041\|last=Barndorff-Nielsen\|first=Ole\|year=1977\|title=Exponentially decreasing distributions for the logarithm of particle size\|journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences\|volume=353\|issue=1674\|pages=401–409\|jstor=79167\|publisher=The Royal Society}}</ref> and is a subclass of the [[generalised hyperbolic distribution]]. The class of NIG distributions is a flexible system of distributions that includes fat-tailed and skewed distributions, and the [[normal distribution]], <math>N(\mu,\sigma^2),</math> arises as a special case by setting <math>\beta=0, \delta=\sigma^2\alpha,</math> and letting <math>\alpha\rightarrow\infty</math>.

			The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If <math>X_1</math> and <math>X_2</math> are [[statistical independence\|independent]] [[random variable]]s that are NIG-distributed with the same values of the parameters <math>\alpha</math> and <math>\beta</math>, but possibly different values of the location and scale parameters, <math>\mu_1</math>, <math>\delta_1</math> and <math>\mu_2,</math> <math>\delta_2</math>, respectively, then <math>X_1 + X_2</math> is NIG-distributed with parameters <math>\alpha, </math> <math>\beta, </math><math>\mu_1+\mu_2</math> and <math>\delta_1 + \delta_2.</math>

			The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion ([[Wiener process]]), <math>W^{(\gamma)}(t)=W(t)+\gamma t</math>, we can define the inverse Gaussian process <math>A_t=\inf\{s>0:W^{(\gamma)}(s)=\delta t\}.</math> Then given a second independent drifting Brownian motion, <math>W^{(\beta)}(t)=\tilde W(t)+\beta t</math>, the normal-inverse Gaussian process is the time-changed process <math>X_t=W^{(\beta)}(A_t)</math>. The process <math>X(t)</math> at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of [[Lévy processes]].

			The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.

			==References==
			<references/>

			{{ProbDistributions\|continuous-infinite}}

			{{DEFAULTSORT:Normal-Inverse Gaussian Distribution}}
			[[Category:Continuous distributions]]
			[[Category:Generalized hyperbolic distributions]]
			[[Category:Probability distributions]]