Arthur–Merlin protocol: Difference between revisions
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{{Probability distribution | | |||
name =Triangular| | |||
type =density| | |||
pdf_image =[[Image:Triangular distribution PMF.png|325px|Plot of the Triangular PMF]]| | |||
cdf_image =[[Image:Triangular distribution CMF.png|325px|Plot of the Triangular CMF]]| | |||
parameters =<math>a:~a\in (-\infty,\infty)</math><br><math>b:~a<b\,</math><br><math>c:~a\le c\le b\,</math>| | |||
support =<math>a \le x \le b \!</math>| | |||
pdf =<math> | |||
\begin{cases} | |||
0 & \mathrm{for\ } x < a, \\ | |||
\frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] | |||
\frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt] | |||
0 & \mathrm{for\ } b < x. | |||
\end{cases} | |||
</math>| | |||
cdf =<math> | |||
\begin{cases} | |||
0 & \mathrm{for\ } x < a, \\[2pt] | |||
\frac{(x-a)^2}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] | |||
1-\frac{(b-x)^2}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt] | |||
1 & \mathrm{for\ } b < x. | |||
\end{cases} | |||
</math>| | |||
mean =<math>\frac{a+b+c}{3}</math>| | |||
median =<math> | |||
\begin{cases} | |||
a+\frac{\sqrt{(b-a)(c-a)}}{\sqrt{2}} & \mathrm{for\ } c \ge \frac{a+b}{2}, \\[6pt] | |||
b-\frac{\sqrt{(b-a)(b-c)}}{\sqrt{2}} & \mathrm{for\ } c \le \frac{a+b}{2}. | |||
\end{cases} | |||
</math>| | |||
mode =<math>c\,</math>| | |||
variance =<math>\frac{a^2+b^2+c^2-ab-ac-bc}{18}</math>| | |||
skewness =<math> | |||
\frac{\sqrt 2 (a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^2\!+\!b^2\!+\!c^2\!-\!ab\!-\!ac\!-\!bc)^\frac{3}{2}} | |||
</math>| | |||
kurtosis =<math>-\frac{3}{5}</math>| | |||
entropy =<math>\frac{1}{2}+\ln\left(\frac{b-a}{2}\right)</math>| | |||
mgf =<math>2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}} | |||
{(b-a)(c-a)(b-c)t^2}</math>| | |||
char =<math>-2\frac{(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}} | |||
{(b-a)(c-a)(b-c)t^2}</math>| | |||
}} | |||
In [[probability theory]] and [[statistics]], the '''triangular distribution''' is a continuous [[probability distribution]] with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. The [[probability density function]] is given by | |||
<math alt="Probability density function for the triangular distribution."> | |||
f(x|a,b,c)= \begin{cases} | |||
0 & \mathrm{for\ } x < a, \\ | |||
\frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] | |||
\frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt] | |||
0 & \mathrm{for\ } b < x, | |||
\end{cases} | |||
</math><br> | |||
whose cases avoid division by zero if ''c'' = ''a'' or ''c'' = ''b''. | |||
==Special cases== | |||
===Two points known=== | |||
The distribution simplifies when ''c'' = ''a'' or ''c'' = ''b''. For example, if ''a'' = 0, ''b'' = 1 and ''c'' = 1, then the equations above become: | |||
:<math> \left.\begin{matrix}f(x) &=& 2x \\[8pt] | |||
F(x) &=& x^2 \end{matrix}\right\} \text{ for } 0 \le x \le 1 </math> | |||
:<math> \begin{align} | |||
E(X) & = \frac{2}{3} \\[8pt] | |||
\mathrm{Var}(X) &= \frac{1}{18} | |||
\end{align} | |||
</math> | |||
===Distribution of mean of two standard uniform variables=== | |||
This distribution for ''a'' = 0, ''b'' = 1 and ''c'' = 0.5 is distribution of ''X'' = (''X''<sub>1</sub> + ''X''<sub>2</sub>)/2, where ''X''<sub>1</sub>, ''X''<sub>2</sub> are two independent random variables with standard [[uniform distribution (continuous)|uniform distribution]]. | |||
:<math> | |||
f(x) = \begin{cases} | |||
4x & \text{for }0 \le x < \frac{1}{2} \\ | |||
4-4x & \text{for }\frac{1}{2} \le x \le 1 | |||
\end{cases} | |||
</math> | |||
:<math> | |||
F(x) = \begin{cases} | |||
2x^2 & \text{for }0 \le x < \frac{1}{2} \\ | |||
1-2(1-x)^2 & \text{for }\frac{1}{2} \le x \le 1 | |||
\end{cases} | |||
</math> | |||
:<math> | |||
\begin{align} | |||
E(X) & = \frac{1}{2} \\[6pt] | |||
\operatorname{Var}(X) & = \frac{1}{24} | |||
\end{align} | |||
</math> | |||
===Distribution of the absolute difference of two standard uniform variables=== | |||
This distribution for ''a'' = 0, ''b'' = 1 and ''c'' = 0 is distribution of ''X'' = |''X''<sub>1</sub> − ''X''<sub>2</sub>|, where ''X''<sub>1</sub>, ''X''<sub>2</sub> are two independent random variables with standard [[uniform distribution (continuous)|uniform distribution]]. | |||
:<math> | |||
\begin{align} | |||
f(x) & = 2 - 2x \text{ for } 0 \le x < 1 \\[6pt] | |||
F(x) & = 2x - x^2 \text{ for } 0 \le x < 1 \\[6pt] | |||
E(X) & = \frac{1}{3} \\[6pt] | |||
\operatorname{Var}(X) & = \frac{1}{18} | |||
\end{align} | |||
</math> | |||
== Generating Triangular-distributed random variates == | |||
Given a random variate ''U'' drawn from the [[Uniform distribution (continuous)|uniform distribution]] in the interval <nowiki>(0, 1)</nowiki>, then the variate | |||
:<math> | |||
\begin{matrix} | |||
\begin{cases} | |||
X = a + \sqrt{U(b-a)(c-a)} & \text{ for } 0 < U < F(c) \\ & \\ | |||
X = b - \sqrt{(1-U)(b-a)(b-c)} & \text{ for } F(c) \le U < 1 | |||
\end{cases} | |||
\end{matrix} | |||
</math><ref>http://www.worldscibooks.com/etextbook/5720/5720_chap1.pdf</ref> | |||
Where F(c) = (c-a)/(b-a) | |||
has a Triangular distribution with parameters a, b and c. This can be obtained from the cumulative distribution function. | |||
==Use of the distribution== | |||
{{see also|Three-point estimation}} | |||
The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). | |||
It is based on a knowledge of the minimum and maximum and an "inspired guess" <ref>http://www.decisionsciences.org/DecisionLine/Vol31/31_3/31_3clas.pdf</ref> as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution. | |||
===Business simulations=== | |||
The triangular distribution is therefore often used in [[Decision making#Decision making in business and management|business decision making]], particularly in [[Simulation#Computer simulation|simulations]]. Generally, when not much is known about the [[Probability distribution|distribution]] of an outcome, (say, only its smallest and largest values) it is possible to use the [[uniform distribution (continuous)|uniform distribution]]. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example under [[Corporate finance#Quantifying uncertainty|corporate finance]]. | |||
===Project management=== | |||
The triangular distribution, along with the [[Beta distribution]], is also widely used in [[project management]] (as an input into [[PERT]] and hence [[critical path method]] (CPM)) to model events which take place within an interval defined by a minimum and maximum value. | |||
===Audio dithering=== | |||
The symmetric triangular distribution is commonly used in [[Dither|audio dithering]], where it is called TPDF (Triangular Probability Density Function). | |||
==See also== | |||
*[[Trapezoidal distribution]] | |||
*[[Thomas Simpson]] | |||
*[[Three-point estimation]] | |||
*[[Five-number summary]] | |||
*[[Seven-number summary]] | |||
*[[Triangular function]] | |||
==External links== | |||
*{{MathWorld|urlname=TriangularDistribution|title=Triangular Distribution}} | |||
*[http://www.decisionsciences.org/DecisionLine/Vol31/31_3/31_3clas.pdf Triangle Distribution], decisionsciences.org | |||
*[http://www.brighton-webs.co.uk/distributions/triangular.htm Triangular Distribution], brighton-webs.co.uk | |||
==References== | |||
{{Reflist}} | |||
{{ProbDistributions|continuous-bounded}} | |||
{{DEFAULTSORT:Triangular Distribution}} | |||
[[Category:Continuous distributions]] | |||
[[Category:Probability distributions]] | |||
Revision as of 21:57, 23 September 2013
Template:Probability distribution
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. The probability density function is given by
whose cases avoid division by zero if c = a or c = b.
Special cases
Two points known
The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the equations above become:
Distribution of mean of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0.5 is distribution of X = (X1 + X2)/2, where X1, X2 are two independent random variables with standard uniform distribution.
Distribution of the absolute difference of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0 is distribution of X = |X1 − X2|, where X1, X2 are two independent random variables with standard uniform distribution.
Generating Triangular-distributed random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
Where F(c) = (c-a)/(b-a)
has a Triangular distribution with parameters a, b and c. This can be obtained from the cumulative distribution function.
Use of the distribution
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The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection).
It is based on a knowledge of the minimum and maximum and an "inspired guess" [2] as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution.
Business simulations
The triangular distribution is therefore often used in business decision making, particularly in simulations. Generally, when not much is known about the distribution of an outcome, (say, only its smallest and largest values) it is possible to use the uniform distribution. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example under corporate finance.
Project management
The triangular distribution, along with the Beta distribution, is also widely used in project management (as an input into PERT and hence critical path method (CPM)) to model events which take place within an interval defined by a minimum and maximum value.
Audio dithering
The symmetric triangular distribution is commonly used in audio dithering, where it is called TPDF (Triangular Probability Density Function).
See also
- Trapezoidal distribution
- Thomas Simpson
- Three-point estimation
- Five-number summary
- Seven-number summary
- Triangular function
External links
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References
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