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  | name      = Logit-normal
  | type      = continuous
  | pdf_image = [[Image:LogitnormalPDF.svg|325px|Plot of the Logitnormal PDF]]
  | cdf_image = [[Image:LogitnormCdfGrid2.svg|325px|Plot of the Logitnormal PDF]]
  | notation  = <math>P( \mathcal{N}(\mu,\,\sigma^2) )</math>
  | parameters= ''σ''<sup>2</sup> > 0 — squared scale (real),<br /> ''μ'' ∈ '''R''' — location
  | support  = ''x'' ∈ (0, 1)
  | pdf      = <math>\frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}}\frac{1}{x (1-x)}</math>
  | cdf      = <math>\frac12\Big[1 + \operatorname{erf}\Big( \frac{\operatorname{logit}(x)-\mu}{\sqrt{2\sigma^2}}\Big)\Big]</math>
  | mean      = no analytical solution
  | median    = <math>P(\mu)\,</math>
  | mode      = no analytical solution
  | variance  = no analytical solution
  | skewness  =  
  | kurtosis  =
  | entropy  =
  | mgf      = no analytical solution
  | char      =
  | fisher    =
  }}
 
In [[probability theory]], a '''logit-normal distribution''' is a [[probability distribution]] of a [[random variable]] whose [[logit]] has a [[normal distribution]]. If ''Y'' is a random variable with a normal distribution, and ''P'' is the [[logistic function]], then ''X''&nbsp;=&nbsp;''P''(''Y'') has a logit-normal distribution; likewise, if ''X'' is logit-normally distributed, then ''Y''&nbsp;=&nbsp;[[logit]](''X'')= log (''X''/(1-''X'')) is normally distributed. It is also known as the '''logistic normal distribution''',<ref>J Atchison and SM Shen. "Logistic-normal distributions: Some properties and uses."  Biometrika, 1980. [http://scholar.google.com/scholar?q=aitchison%20logistic%20normal Google Scholar link]</ref> which often refers to a multinomial logit version (e.g. <ref>http://people.csail.mit.edu/tomasz/papers/huang_hln_tech_report_2006.pdf</ref> <ref>Peter Hoff, 2003.  [http://www.stat.washington.edu/hoff/Preprints/nhed.ps Link]</ref> <ref>http://www.springerreference.com/docs/html/chapterdbid/205424.html</ref> <ref>http://brenocon.com/blog/2011/05/log-normal-and-logistic-normal-terminology/</ref>).
 
A variable might be modeled as logit-normal if it is a proportion, which is bounded by zero and one, and where values of zero and one never occur.
 
== Characterization ==
 
=== Probability density function ===
The [[probability density function]] of a logit-normal distribution is:
 
: <math>f_X(x;\mu,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}} \frac{1}{x (1-x)}, \quad x \in (0, 1) </math>
 
where ''μ'' and ''σ'' are the [[mean]] and [[standard deviation]] of the variable’s [[logit]] (by definition, the variable’s logit is normally distributed).
 
[[Image:LogitnormDensityGrid.svg|630px|Plot of the Logitnormal PDF for various combinations of ''μ'' (facets) and ''σ'' (colors)]]
 
The density obtained by changing the sign of ''μ'' is symmetrical, in that it is equal to f(1-x;-''μ'',''σ''), shifting the mode to the other side of 0.5 (the midpoint of the (0,1) interval).
 
=== Moments ===
The moments of the logit-normal distribution have no analytic solution. However, they can be
estimated by [[numerical integration]].
 
=== Mode ===
When the derivative of the density equals 0 then the location of the mode x satisfies the following equation:
 
:<math>\operatorname{logit}(x) = \sigma^2(2x-1)+\mu .</math>
 
== See also ==
* [[Beta distribution]] and [[Kumaraswamy distribution]], other two-parameter distributions on a bounded interval with similar shapes
 
== Further reading ==
* Frederic, P. & Lad, F. (2008) [http://www.economia.unimore.it/frederic_patrizio/Hidden/Papers/2008%20Frederic-Lad%20Comm%20Stat%20Revised.pdf Two Moments of the Logitnormal Distribution.] ''Communications in Statistics-Simulation and Computation''. 37: 1263-1269
* {{cite JSTOR|2528553}}
{{reflist}}
 
==External links==
* [http://logitnorm.r-forge.r-project.org/ logitnorm package] for [[R (programming language)|R]]
 
{{ProbDistributions|continuous-bounded}}
 
{{DEFAULTSORT:Logit-Normal Distribution}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Latest revision as of 21:53, 27 July 2014

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The nature of this psycho-social poison from televised violence was highlighted for me by an experience in Tasmania in the late 1970’s that I have retold many instances. I had arranged to purchase the22 calibre rifle that I nonetheless personal as a “licensed weapon” from Janet Mollison (Bill Mollison’s oldest daughter who was my very own age). On the time, Janet lived in a flat above the Salvo’s Op shop in Elizabeth St, North Hobart. I had arranged to choose up the rifle on a Saturday evening. The transaction was simple with no paper work being required for possession of rim-fireplace rifles in those occasions.

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Ancient Rome was a a lot totally different world than we live in at this time. Sure comparable threats still exist right this moment between our culture and theirs. But civilization has safeguarded us much more than those living two thousand years ago. Our two cultures additionally share a love and appreciation for canine. Romans cherished their animals, as can be seen by their religions. Canines have been usually symbolized of their spiritual practices. However dogs were much more in the Roman tradition than just non secular symbols. In every single day life they served many useful functions. Their two primary features were to hunt and protect.

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