Editing Triangular distribution

{{short description|Probability distribution}}
{{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 & \text{for } x < a, \\
    \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x < c, \\[4pt]
    \frac{2}{b-a}             & \text{for } x = c, \\[4pt]
    \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c < x \le b, \\[4pt]
    0 & \text{for } b < x.
 \end{cases}
              </math>|
  cdf        =<math>
  \begin{cases}
    0 & \text{for } x \leq a, \\[2pt]
    \frac{(x-a)^2}{(b-a)(c-a)} & \text{for } a < x \leq c, \\[4pt]
    1-\frac{(b-x)^2}{(b-a)(b-c)} & \text{for } c < x < b, \\[4pt]
    1 & \text{for } b \leq x.
  \end{cases}
              </math>|
  mean       =<math>\frac{a+b+c}{3}</math>|
  median     =<math>
  \begin{cases}
    a+\sqrt{\frac{(b-a)(c-a)}{2}} & \text{for } c \ge \frac{a+b}{2}, \\[6pt]
    b-\sqrt{\frac{(b-a)(b-c)}{2}} & \text{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''&nbsp;<&nbsp;''b'' and ''a''&nbsp;≤&nbsp;''c''&nbsp;≤&nbsp;''b''.

==Special cases==
===Mode at a bound===
The distribution simplifies when ''c''&nbsp;=&nbsp;''a'' or ''c''&nbsp;=&nbsp;''b''.  For example, if ''a''&nbsp;=&nbsp;0, ''b''&nbsp;=&nbsp;1 and ''c''&nbsp;=&nbsp;1, then the [[Probability density function|PDF]] and [[Cumulative distribution function|CDF]] become:

:<math> \left.\begin{array}{rl} f(x) &= 2x \\[8pt]
F(x) &= x^2 \end{array}\right\} \text{ for } 0 \le x \le 1 </math>

:<math> \begin{align}
  \operatorname E(X) & = \frac{2}{3} \\[8pt]
  \operatorname{Var}(X) &= \frac{1}{18}
\end{align} </math>

====Distribution of the absolute difference of two standard uniform variables====
This distribution for ''a''&nbsp;=&nbsp;0, ''b''&nbsp;=&nbsp;1 and ''c''&nbsp;=&nbsp;0 is the distribution of ''X''&nbsp;=&nbsp;|''X''<sub>1</sub>&nbsp;&minus;&nbsp;''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>

===Symmetric triangular distribution===
The symmetric case arises when ''c'' = (''a'' + ''b'') / 2.
In this case, an alternate form of the distribution function is:

:<math> \begin{align}
  f(x) &= \frac{(b-c)-|c-x|}{(b-c)^2} \\[6pt]
\end{align} </math>

====Distribution of the mean of two standard uniform variables====

This distribution for ''a''&nbsp;=&nbsp;0, ''b''&nbsp;=&nbsp;1 and ''c''&nbsp;=&nbsp;0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of ''X''&nbsp;=&nbsp;(''X''<sub>1</sub>&nbsp;+&nbsp;''X''<sub>2</sub>)&nbsp;/&nbsp;2, where ''X''<sub>1</sub>, ''X''<sub>2</sub> are two independent random variables with standard [[uniform distribution (continuous)|uniform distribution]] in [0, 1].<ref name="KD">{{Cite book |last=Kotz |first=Samuel |url=https://books.google.com/books?id=JO7ICgAAQBAJ&dq=chapter%201%20dig%20out%20suitable%20substitutes%20of%20the%20beta%20distribution%20one%20of%20our%20goals&pg=PA3 |title=Beyond Beta: Other Continuous Families Of Distributions With Bounded Support And Applications |last2=Dorp |first2=Johan Rene Van |date=2004-12-08 |publisher=World Scientific |isbn=978-981-4481-24-3 |language=en}}</ref> It is the case of the [[Bates distribution]] for two variables.

:<math>
  f(x) = \begin{cases}
  4x   & \text{for }0 \le x < \frac{1}{2}   \\
  4(1-x) & \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} \\
  2x^2-(2x-1)^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>

==Generating random variates==
Given a random variate ''U'' drawn from the [[Uniform distribution (continuous)|uniform distribution]] in the interval <nowiki>(0,&nbsp;1)</nowiki>, then the variate

:<math>
X = \begin{cases}
a + \sqrt{U(b-a)(c-a)} & \text{ for } 0 < U < F(c) \\ & \\
b - \sqrt{(1-U)(b-a)(b-c)} & \text{ for } F(c) \le U < 1
\end{cases}
</math><ref>{{cite web |url=http://www.asianscientist.com/books/wp-content/uploads/2013/06/5720_chap1.pdf |title=Archived copy |website=www.asianscientist.com |access-date=12 January 2022 |archive-url=https://web.archive.org/web/20140407075018/http://www.asianscientist.com/books/wp-content/uploads/2013/06/5720_chap1.pdf |archive-date=7 April 2014 |url-status=dead}}</ref>

where <math>F(c) = (c-a)/(b-a)</math>, has a triangular distribution with parameters <math>a, b</math> and <math>c</math>. 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>{{cite web |url=http://www.decisionsciences.org/DecisionLine/Vol31/31_3/31_3clas.pdf |title=Archived copy |access-date=2006-09-23 |url-status=dead |archive-url=https://web.archive.org/web/20060923225843/http://www.decisionsciences.org/DecisionLine/Vol31/31_3/31_3clas.pdf |archive-date=2006-09-23 }}</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 [[PERT 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]]
*[[Central limit theorem]] — The triangle distribution often occurs as a result of adding two uniform random variables together. In other words, the triangle distribution is often (not always) the result of the first iteration of the central limit theorem summing process (i.e. <math display="inline">n = 2</math>). In this sense, the triangle distribution can occasionally occur naturally. If this process of summing together more random variables continues (i.e. <math display="inline">n \geq 3</math>), then the distribution will become increasingly bell-shaped.
*[[Irwin–Hall distribution]] — Using an Irwin–Hall distribution is an easy way to generate a triangle distribution.
*[[Bates distribution]] — Similar to the Irwin–Hall distribution, but with the values rescaled back into the 0 to 1 range. Useful for computation of a triangle distribution which can subsequently be rescaled and shifted to create other triangle distributions outside of the 0 to 1 range.

==References==
{{Reflist}}

==External links==
*{{MathWorld|urlname=TriangularDistribution|title=Triangular Distribution}}
*[https://web.archive.org/web/20060923225843/http://www.decisionsciences.org/DecisionLine/Vol31/31_3/31_3clas.pdf Triangle Distribution], decisionsciences.org
*[https://web.archive.org/web/20130318003944/http://www.brighton-webs.co.uk/distributions/triangular.htm Triangular Distribution], brighton-webs.co.uk
*[https://math.stackexchange.com/questions/4271314/what-is-the-proof-for-variance-of-triangular-distribution/4273147#4273147 Proof for the variance of triangular distribution], math.stackexchange.com

{{ProbDistributions|continuous-bounded}}

{{DEFAULTSORT:Triangular Distribution}}
[[Category:Continuous distributions]]