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{{Refimprove|date=September 2009}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


In [[statistics]], a '''truncated distribution''' is a [[conditional distribution]] that results from restricting the domain of some other [[probability distribution]]. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range. For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the school's cutoff dates if only a direct approach to the school were used to obtain information.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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Where sampling is such as to retain knowledge of items that fall outside the required range, without recording the actual values, this is known as [[Censoring (statistics)|censoring]], as opposed to the [[Truncation (statistics)|truncation]] here.<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms''. OUP. ISBN 0-19-020613-9 {{Please check ISBN|reason=Check digit (9) does not correspond to calculated figure.}}</ref>
Registered users will be able to choose between the following three rendering modes:


==Definition==
'''MathML'''
{{Probability distribution|
:<math forcemathmode="mathml">E=mc^2</math>
  name      =Truncated Distribution|
  type      =density|
  pdf_image  =<!-- Deleted image removed: [[File:Truncation.gif|Probability density function for a truncated standard normal distribution, truncated at -1 and 1 {{Deletable image-caption|date=May 2012}}]] --><br /><small>The red line is a truncated standard normal distribution, truncated at -1 and 1</small>|
  cdf_image  =[[File:Truncation CDF.gif|Cumulative distribution function for a trucated standard normal distribution truncated at -1 and 1]] |
  parameters = The parameters of <math> f(x) </math>, plus <math> a </math> and <math> b </math>|
  support    =<math>x \in (a,b]</math>|
  pdf        =<math>\frac{g(x)}{F(b)-F(a)} </math>|
  cdf        =<math>\frac{\int_a^xg(t)dt}{F(b)-F(a)} </math>|
  mean      =<math>\frac{\int_a^b x g(x) dx}{F(b)-F(a)} </math>|
  median    =|
  mode      =|
  variance  =|
  skewness  =|
  kurtosis  =|
  entropy    =|
  mgf        =|
  char      =|


}}
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The following discussion is in terms of a random variable having a [[continuous distribution]] although the same ideas apply to [[discrete distribution]]s. Similarly, the discussion assumes that truncation is to a semi-open interval ''y'' ∈ (''a,b''] but other possibilities can be handled straightforwardly.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Suppose we have a random variable, <math> X </math> that is distributed according to some probability density function, <math> f(x) </math>, with cumulative distribution function <math> F(x) </math> both of which have infinite [[Support (mathematics)|support]].  Suppose we wish to know the probability density of the random variable after restricting the support to be between two constants so that the support,  <math> y = (a,b] </math>. That is to say, suppose we wish to know how <math> X </math> is distributed given <math> a < X \leq b </math>.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


:<math>f(x|a < X \leq b) = \frac{g(x)}{F(b)-F(a)} = Tr(x)</math>
==Demos==


where <math>g(x) = f(x)</math> for all <math> a <x \leq b </math> and <math> g(x) = 0 </math> everywhere else. Notice that <math>Tr(x)</math> has the same support as  <math>g(x)</math>.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


There is, unfortunately, an ambiguity about the term Truncated Distribution. When one refers to a truncated distribution one could be referring to <math> g(x) </math> where one has removed the parts from the distribution <math> f(x) </math> but not scaled up the distribution, or one could be referring to the <math> Tr(x)</math>.  In general, <math> g(x) </math> is not a probability density function since it does not integrate to one, whereas <math> Tr(x)</math> is a probability density function.  In this article, a truncated distribution refers to <math> Tr(x)</math>


Notice that in fact <math>f(x|a < X \leq b)</math> is a distribution:
* accessibility:
:<math>\int_{a}^{b} f(x|a < X \leq b)dx = \frac{1}{F(b)-F(a)} \int_{a}^{b} g(x) dx = 1 </math>.
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Truncated distributions need not have parts removed from the top and bottom. A truncated distribution where just the bottom of the distribution has been removed is as follows:
==Test pages ==


:<math>f(x|X>y) = \frac{g(x)}{1-F(y)}</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


where <math>g(x) = f(x)</math> for all <math> y < x </math> and <math> g(x) = 0 </math> everywhere else, and <math>F(x)</math> is the [[cumulative distribution function]].
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
A truncated distribution where the top of the distribution has been removed is as follows:
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<math>f(x|X \leq y) = \frac{g(x)}{F(y)}</math>
 
where <math>g(x) = f(x)</math> for all <math> x \leq y </math> and <math> g(x) = 0 </math> everywhere else, and <math>F(x)</math> is the [[cumulative distribution function]].
 
== Expectation of truncated random variable ==
Suppose we wish to find the expected value of a random variable distributed according to the density <math> f(x) </math> and a cumulative distribution of <math> F(x) </math> given that the random variable, <math> X </math>, is greater than some known value <math> y </math>. The expectation of a truncated random variable is thus:
 
<math> E(X|X>y) = \frac{\int_y^\infty x g(x) dx}{1 - F(y)} </math>
 
where again <math> g(x) </math> is <math>g(x) = f(x)</math> for all <math> y < x </math> and <math> g(x) = 0 </math> everywhere else.
 
Letting <math> a </math> and <math> b </math> be the lower and upper limits respectively of support for <math>f(x)</math> (i.e. the original density) properties of <math> E(u(X)|X>y) </math> where <math>u(X)</math> is some continuous function of <math> X </math> with a continuous derivative and where <math> f(x) </math> is assumed continuous include:
 
(i)  <math> \lim_{y \to a} E(u(X)|X>y) = E(u(X)) </math>
 
(ii)  <math> \lim_{y \to b} E(u(X)|X>y) = u(b) </math>
 
(iii)  <math> \frac{\partial}{\partial y}[E(u(X)|X>y)] = \frac{f(y)}{1-F(y)}[E(u(X)|X>y) - u(y)] </math>
 
(iv)  <math> \lim_{y \to a}\frac{\partial}{\partial y}[E(u(X)|X>y)] = f(a)[E(u(X)) - u(a)] </math>
 
(v)  <math> \lim_{y \to b}\frac{\partial}{\partial y}[E(u(X)|X>y)] = \frac{1}{2}u'(b) </math>
 
Provided that the limits exist, that is: <math> \lim_{y \to c} u'(y) = u'(c) </math>, <math> \lim_{y \to c} u(y) = u(c) </math> and <math>\lim_{y \to c} f(y) = f(c) </math> where <math> c </math> represents either <math>a</math> or <math> b</math>.
 
==Examples==
 
The [[truncated normal distribution]] is an important example.<ref> Johnson, N.L., Kotz, S., Balakrishnan, N. (1994) ''Continuous Univariate Distributions, Volume 1'', Wiley. ISBN 0-471-58495-9 (Section 10.1)</ref>
 
The [[Tobit model]] employs truncated distributions.
 
== Random truncation ==
Suppose we have the following set up: a truncation value, <math>t</math>, is selected at random from a density, <math>g(t)</math>, but this value is not observed. Then a value, <math>x</math>, is selected at random from the truncated distribution, <math>f(x|t)=Tr(x)</math>.  Suppose we observe <math>x</math> and wish to update our belief about the density of <math>t</math> given the observation.
 
First, by definition:
 
:<math>f(x)=\int_{x}^{\infty} f(x|t)g(t)dt </math>, and
:<math>F(a)=\int_{-\infty}^a[\int_{x}^{\infty} f(x|t)g(t)dt]dx .</math>
 
Notice that <math>t</math> must be greater than <math>x</math>, hence when we integrate over <math>t</math>, we set a lower bound of <math>x</math>. The functions <math>f(x)</math> and <math>F(x)</math> are the unconditional density and unconditional cumulative distribution function, respectively.
 
By [[Bayes' rule]],
 
:<math>g(t|x)= \frac{f(x|t)g(t)}{f(x)} ,</math>
 
which expands to
 
:<math>g(t|x) = \frac{f(x|t)g(t)}{\int_{x}^{\infty} f(x|t)g(t)dt} .</math>
 
=== Two uniform distributions (example) ===
Suppose we know that ''t'' is uniformly distributed from [0,''T''] and ''x''|''t'' is distributed uniformly on [0,''t''].  Let ''g''(''t'') and ''f''(''x''|''t'') be the densities that describe ''t'' and ''x'' respectively.  Suppose we observe a value of ''x'' and wish to know the distribution of ''t'' given that value of ''x''.
 
:<math>g(t|x) =\frac{f(x|t)g(t)}{f(x)} = \frac{1}{t(\ln(t) - \ln(x))} \quad \text{for all } t > x .</math>
 
==See also==
 
*[[Truncated mean]]
 
==References==
<references/>
 
 
[[Category:Theory of probability distributions]]
[[Category:Types of probability distributions]]
 
[[fr:Loi tronquée]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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