Triangular distribution

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Triangular
Probability density function
Plot of the Triangular PMF
Cumulative distribution function
Plot of the Triangular CMF
Parameters a:~a\in (-\infty,\infty)
b:~b>a\,
c:~a\le c\le b\,
Support a \le x \le b \!
Probability density function (pdf) \left\{ \begin{matrix} \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \le c \\ & \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c \le x \le b \end{matrix} \right.
Cumulative distribution function (cdf) \left\{ \begin{matrix} \frac{(x-a)^2}{(b-a)(c-a)} & \mathrm{for\ } a \le x \le c \\ & \\ 1-\frac{(b-x)^2}{(b-a)(b-c)} & \mathrm{for\ } c \le x \le b \end{matrix} \right.
Mean \frac{a+b+c}{3}
Median \left\{ \begin{matrix} a+\frac{\sqrt{(b-a)(c-a)}}{\sqrt{2}} & \mathrm{for\ } c\!\ge\!\frac{b\!-\!a}{2}\\ & \\ b-\frac{\sqrt{(b-a)(b-c)}}{\sqrt{2}} & \mathrm{for\ } c\!\le\!\frac{b\!-\!a}{2} \end{matrix} \right.
Mode c\,
Variance \frac{a^2+b^2+c^2-ab-ac-bc}{18}
Skewness \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}}
Excess kurtosis -\frac{3}{5}
Entropy \frac{1}{2}+\ln\left(\frac{b-a}{2}\right)
Moment-generating function (mgf) 2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}} {(b-a)(c-a)(b-c)t^2}
Characteristic function -2\frac{(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}} {(b-a)(c-a)(b-c)t^2}

In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b.

f(x|a,b,c)=\left\{ \begin{matrix} \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \le c \\ & \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c \le x \le b \end{matrix} \right.

Contents

The distribution simplifies when c=a or c=b. For example, if a=0, b=1 and c=1, then the equations above become:

\left.\begin{matrix}f(x) &=& 2x \\ \\ F(x) &=& x^2 \end{matrix}\right\} \mathrm{for\ } 0 \le x \le 1
\begin{matrix} E(X) &=& \frac{2}{3} \\ & & \\ \mathrm{Var}(X) &=& \frac{1}{18} \end{matrix}

This distribution for a=0, b=1 and c=0.5 is distribution of X = \frac{X_1+X_2}{2}, where X1,X2 are two random variables with standard uniform distribution.

f(x)=\left\{\begin{matrix} 4x & \mathrm{for\ }0 \le x < \frac{1}{2} \\ \\ 4-4x & \mathrm{for\ }\frac{1}{2} \le x \le 1 \end{matrix}\right.
F(x)=\left\{\begin{matrix} 2x^2 & \mathrm{for\ }0 \le x < \frac{1}{2} \\ \\ 1-2(1-x)^2 & \mathrm{for\ }\frac{1}{2} \le x \le 1 \end{matrix}\right.
\begin{matrix} E(X) &=& \frac{1}{2} \\ \\ \mathrm{Var}(X) &=& \frac{1}{24} \end{matrix}


This distribution for a=0, b=1 and c=0 is distribution of X = | X1 − X2 | , where X1,X2 are two random variables with standard uniform distribution.


\begin{matrix} f(x)&=& 2 - 2x \qquad \mathrm{for\ } 0 \le x < 1 \\ \\ F(x) &=& 2x - x^2 \qquad \mathrm{for\ } 0 \le x < 1 \\ \\ \end{matrix}


\begin{matrix} E(X) &=& \frac{1}{3} \\ \\ \mathrm{Var}(X) &=& \frac{1}{18} \end{matrix}

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" as to the modal value.

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.

The Triangular distribution, along with the Beta distribution, is also widely used in project management (as an input into PERT and hence CPM) to model events which take place within an interval defined by a minimum and maximum value.

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