en>Faizan: clean up and fixes, typo(s) fixed: , → , , of of → of using AWB

2014-02-25T16:11:20Z

clean up and fixes, typo(s) fixed: , → , , of of → of using AWB

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	~~{{Probability distribution~~		Not much to write about me at all.<br>Hurrey Im here and a part of this community.<br>I really wish I am useful in some way here.<br><br>Feel free to surf to my homepage; [http://officialweightlosshelp.com/?q=node/818 backup plugin]
	~~\| type = density~~
	~~\| pdf_image =~~
	~~\| cdf_image =~~
	~~\| notation = Tukey(''λ'')~~
	~~\| parameters = ''λ'' ∈ '''R''' — [[shape parameter]]~~
	~~\| support = ''x'' ∈ [−1/''λ'', 1/''λ''] for ''λ'' > 0,<br/>''x'' ∈ '''R''' for ''λ'' ≤ 0~~
	~~\| pdf = <math>(Q(p;\lambda)\,,Q'(p;\lambda)^{-1}),\, 0\leq\,p\,\leq\,1</math>~~
	~~\| cdf = <math>(e^{-x}+1)^{-1},\,\,\lambda\,=\,0</math>~~
	~~\| mean = <math>0,\,\,\lambda > -1</math>~~
	~~\| median = 0~~
	~~\| mode = 0~~
	\| variance = <math>\frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda}-\frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg),\,\,\lambda > -1/2</math><br/><math>\frac{ \pi^{2} }{ 3 },\,\,\lambda\,=\,0</math>
	~~\| skewness = <math>0,\,\,\lambda > -1/3</math>~~
	\| kurtosis = <math>\frac{(2\lambda+1)^2}{2(4\lambda+1)} \frac{ g_2^2\big(3g_2^2-4g_1g_3+g_4\big)}{g_4\big(g_1^2-g_2\big)^2} - 3,</math><br/><math> 1.2,\,\,\lambda\,=\,0,</math> where ''g''<sub>''k''</sub> = Γ(''kλ''+1) and ''λ'' > -1/4.
	\| entropy = <math>h(\lambda) = \int_0^1 \log (Q'(p;\lambda))\,dp</math><ref>{{Citation \|last1=Vasicek \|first1=Oldrich \|year=1976 \|title=A Test for Normality Based on Sample Entropy \|journal=Journal of the Royal Statistical Society, Series B \|volume=38 \|issue=1 \|pages=54–59 \|postscript=. }}</ref>
	~~\| mgf =~~
	\| cf = <math>\phi(t;\lambda) = \int_0^1 \exp (\,i t\,Q(p;\lambda))\,dp</math><ref>{{Citation \|last1=Shaw \|first1=W. T. \|last2=McCabe \|first2=J. \|year=2009 \|title=Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space \|journal=Eprint-arXiv:0903,1592 }}</ref>
	}}
	~~Formalized by [[John Tukey]], the '''Tukey lambda distribution''' is a continuous probability distribution defined in terms of its [[quantile function]]. It is typically used~~ to ~~identify an appropriate distribution (see the comments below) and not used in [[statistical model]]s directly.~~

	The Tukey lambda distribution has a single [[shape parameter]] λ. As with other probability distributions, the Tukey lambda distribution can be transformed with a [[location parameter]], μ, and a [[scale parameter]], σ. ~~Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.~~

	~~==Quantile function==~~

	For the standard form of the Tukey lambda distribution, the quantile function, Q(p), (i.e. the inverse of the [[cumulative distribution function]]) and the quantile density function (i.e. the derivative of the quantile function) are

	:<~~math~~>
	~~Q\left(p;\lambda\right) =~~
	~~\begin{cases}~~
	~~\frac{ 1 }{ \lambda } \left[p^\lambda - (1 - p)^\lambda\right], & \mbox{if } \lambda \ne 0 \\~~
	~~\log(\frac{p}{1-p}), & \mbox{if } \lambda = 0,~~
	~~\end{cases}</math>~~
	~~:<math>Q'\left(p;\lambda\right) = p^{(\lambda-1)} + \left(1-p\right)^{(\lambda-1)}.</math>~~

	~~The [[probability density function]] (pdf)~~ and ~~[[cumulative distribution function]] (cdf) are both computed numerically, as the Tukey lambda distribution does not have~~ a simple, closed form for any values of the parameters except ''λ'' = 0 (see [[logistic distribution]]). However, the pdf can be expressed in parametric form, for all values of ''λ'', in terms of the quantile function and the reciprocal of the quantile density function.

	~~==Moments==~~
	~~The Tukey lambda distribution is symmetric around zero, therefore the expected value~~ of this ~~distribution is equal to zero~~. ~~The variance exists for {{nowrap\|''λ'' > −½}} and is given by the formula (except when ''λ'' = 0)~~
	: <~~math>~~
	~~\operatorname{Var}[X] = \frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda} - \frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg).~~
	~~</math>~~

	~~More generally, the ''n''-th order moment is finite when {{nowrap\|''λ''~~ > ~~−1/''n''}} and is expressed~~ in ~~terms of the [[beta function]] ''Β''(''x'',''y'') (except when ''λ'' = 0) :~~
	~~: <math>~~
	~~\mu_n = \operatorname{E}[X^n] = \frac{1}{\lambda^n} \sum_{k=0}^n (-1)^k {n \choose k}\, \Beta(\lambda k+1,\, \lambda(n-k)+1 )~~.
	<~~/math>~~

	~~Note that due to symmetry of the density function, all moments of odd orders are equal to zero.~~

	~~==Comments==~~
	~~The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,~~
	~~{\| class="wikitable"~~
	\|-
	~~\| ''λ'' = −1~~
	~~\| approx. [[Cauchy distribution\|Cauchy]] ''C''(0,''π'')~~
	\|-
	~~\| ''λ'' = 0~~
	~~\| exactly [[logistic distribution\|logistic]]~~
	\|-
	~~\| ''λ'' = 0.14~~
	~~\| approx. [[normal distribution\|normal]] ''N''(0, 2.142)~~
	\|-
	~~\| ''λ'' = 0.5~~
	~~\| strictly [[concave function\|concave]] (<math~~>~~\cap~~<~~/math~~>~~-shaped)~~
	\|-
	~~\| ''λ'' = 1~~
	~~\| exactly [[continuous uniform distribution\|uniform]] ''U''(−1, 1)~~
	\|-
	~~\| ''λ'' = 2~~
	~~\| exactly [[continuous uniform distribution\|uniform]] ''U''(−½, ½)~~
	\|}

	~~The most common use of this distribution is~~ to generate a Tukey lambda [[PPCC plot]] of a [[data set]]. Based on the PPCC plot, an appropriate [[statistical model\|model]] for the data is suggested. For example, if the maximum [[correlation]] occurs for a value of ''λ'' at or near 0.14, then the data can be modeled with a normal distribution. Values of ''λ'' less than this imply a heavy-tailed distribution (with −1 approximating a Cauchy). That is, as the optimal value of lambda goes from 0.14 to ~~−1, increasingly heavy tails are implied. Similarly, as the optimal value of ''λ'' becomes greater than 0.14, shorter tails are implied.~~

	Since the Tukey lambda distribution is a [[reflection symmetry\|symmetric]] distribution, the use of the Tukey lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. A [[histogram]] of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.<ref>{{Citation\| title=Some Properties of the Range in Samples from Tukey's Symmetric Lambda Distributions\| first1=Brian L. \|last1=Joiner\|first2=Joan R. \|last2=Rosenblatt\| journal=Journal of the American Statistical Association\| volume=66 \|issue=334 \|year=1971\| pages=394–~~399\| doi=10.2307/2283943\| jstor=2283943}}</ref>~~

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	*[http://~~www.itl.nist~~.~~gov~~/~~div898~~/~~handbook/eda/section3/eda366f.htm Tukey-Lambda Distribution]~~

	~~{{NIST-PD}}~~
	~~{{ProbDistributions\|continuous-variable}}~~

	~~[[Category:Continuous distributions]]~~
	~~[[Category:Probability distributions with non-finite variance]]~~
	~~[[Category:Probability distributions]~~]

89.70.239.182: /* Airy stress function */

2014-01-16T22:35:12Z

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Stress functions - Revision history

en>Faizan: clean up and fixes, typo(s) fixed: , → , , of of → of using AWB

89.70.239.182: /* Airy stress function */

en>Helpful Pixie Bot: ISBNs (Build KH)