formulasearchengine - User contributions [en]

Constant elasticity of variance model

2014-03-10T14:51:15Z

182.74.99.169: /* Dynamics */

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Single-ended primary-inductor converter

2014-01-23T05:20:42Z

182.74.116.150: /* Circuit operation */

{{Refimprove|date=September 2009}}

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.

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>

==Definition==
{{Probability distribution|
name =Truncated Distribution|
type =density|
| pdf_image = [[File:tnormPDF.png|300px|thumbnail|right|Probability density function for the truncated normal distribution for different sets of parameters. In all cases, ''a'' = −10 and ''b'' = 10. For the black: ''μ'' = −8, ''σ'' = 2; blue: ''μ'' = 0, ''σ'' = 2; red: ''μ'' = 9, ''σ'' = 10; orange: ''μ'' = 0, ''σ'' = 10.]] |
|
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 =|

}}

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.

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>.

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

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>.

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:
:<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>.

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:

:<math>f(x|X>y) = \frac{g(x)}{1-F(y)}</math>

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]].

A truncated distribution where the top of the distribution has been removed is as follows:

:<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 \left[\int_{x}^{\infty} f(x|t)g(t)dt \right]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]]

Differential amplifier

2014-01-17T10:52:31Z

182.74.42.2: /* Biasing */

{{Even polygon db|Even polygon stat table|p10}}
[[File:Gonbad-e Qabus.JPG|thumb|[[Gonbad-e Qabus (tower)|Gonbad-e Qabus]], the tallest pure brick tower in the world, is built on a decagonal plan.]]
In [[geometry]], a '''decagon''' is any [[polygon]] with ten sides and ten [[angle]]s. A [[regular polygon|regular]] decagon has all sides of equal length and each internal angle equal to 144°. Its [[Schläfli symbol]] is {10}.

==Regular decagon==
The [[area]] of a regular decagon is: (with ''t'' = edge length)
:<math>A = \frac{5}{2}t^2 \cot \frac{\pi}{10} = \frac{5t^2}{2} \sqrt{5+2\sqrt{5}} \simeq 7.694208843 t^2.</math>

An alternative formula is <math>\scriptstyle A\,=\,2.5dt</math> where ''d'' is the distance between parallel sides, or the height when the decagon stands on one side as base. 
By simple trigonometry <math>\scriptstyle d\,=\,2t(\cos{54^\circ}\,+\,\cos{18^\circ})</math>.

==Sides==
The side of a regular decagon inscribed in a unit circle is <math>\tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\phi}</math>, where ''ϕ'' is the [[golden ratio]], <math>\tfrac{1+\sqrt{5}}{2}</math>.

===Construction===
A regular decagon is [[constructible polygon|constructible]] using [[compass and straightedge]]:

[[File:Regular Decagon Inscribed in a Circle.gif|Construction of a regular decagon]]

An alternative (but similar) method is as follows:
#Construct a pentagon in a circle by one of the methods shown in [[Pentagon#Construction_of_a_regular_pentagon|constructing a pentagon]].
#Extend a line from each vertex of the pentagon through the center of the [[circle]] to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon.
#The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.

==Related figures==
There is one regular [[star polygon]], the [[decagram (geometry)|decagram]] {10/3}, using the same points, but connecting every third points. There are also two compounds: {10/4} is reduced to 2{5/2} as two [[pentagram]]s, and {10/2} is reduced to 2{5} as two [[pentagon]]s.

{| class=wikitable width=360
|- align=center
|[[File:Truncated pentagon.png|120px]] A [[truncation (geometry)|truncated]] regular pentagon
|[[File:Decagram_10_3.png|120px]] {10/3} [[Decagram (geometry)|Decagram]]
|[[Image:Decagram 10 2.png|120px]] {10/2} or 2{5}
|[[Image:Decagram 10 4.png|120px]] {10/4} or 2{5/2}
|}

===Petrie polygons===
The regular decagon is the [[Petrie polygon]] for many higher dimensional polytopes, shown in these skew [[orthogonal projection]]s in various [[Coxeter plane]]s:

{| class=wikitable width=450
|- align=center valign=top
!valign=center|A9
|[[File:9-simplex_t0.svg|100px]] [[9-simplex]]
|[[File:9-simplex_t1.svg|100px]] [[Rectified 9-simplex]]
|[[File:9-simplex_t2.svg|100px]] [[Birectified 9-simplex]]
|[[File:9-simplex_t3.svg|100px]] [[Trirectified 9-simplex]]
|[[File:9-simplex_t4.svg|100px]] [[Quadrirectified 9-simplex]]
|- align=center valign=top
!valign=center|BC5
|[[File:5-cube_t4.svg|100px]] [[5-orthoplex]]
|[[File:5-cube_t3.svg|100px]] [[Rectified 5-orthoplex]]
|[[File:5-cube_t2.svg|100px]] [[Birectified 5-cube]]
|[[File:5-cube_t1.svg|100px]] [[Rectified 5-cube]]
|[[File:5-cube_t0.svg|100px]] [[5-cube]]
|- align=center valign=top
!valign=center|D6
|[[File:6-cube_t5_B5.svg|100px]] [[5-orthoplex|t1(431)]]
|[[File:6-cube_t4_B5.svg|100px]] [[Rectified 5-orthoplex|t3(131)]]
|[[File:6-cube_t3_B5.svg|100px]] [[Birectified 5-orthoplex|t2(131)]]
|[[File:6-demicube_t1_D6.svg|100px]] [[Rectified 6-demicube|t1(131)]]
|[[File:6-demicube_t0_D6.svg|100px]] [[6-demicube]] (131)
|- align=center valign=top
!valign=center|H3
|[[File:Dodecahedron petrie.png|100px]] [[Dodecahedron]]
|[[File:Icosahedron petrie.png|100px]] [[Icosahedron]]
|[[File:Dodecahedron t1 H3.png|100px]] [[Icosidodecahedron]]
|}

==See also==
*[[decagonal number]]
*[[Gambrel]]
*[[Golden ratio]]

==External links==
*{{MathWorld |urlname=Decagon |title=Decagon}}
*[http://www.mathopenref.com/decagon.html Definition and properties of a decagon] With interactive animation

{{Polygons}}

[[Category:Polygons]]