Triangular distribution

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Triangular
Probability density function
Cumulative distribution function
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.

$f(x|a,b,c)=\left\{ \begin{matrix} \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \le c \\ & \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c \le x \le b \\ & \\ 0 & \mathrm{for\ any\ other\ case} \end{matrix} \right.$

1 Special cases
2 Use of the distribution
3 See also
4 External links

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:

$\left.\begin{matrix}f(x) &=& 2x \\ \\ F(x) &=& x^2 \end{matrix}\right\} \mathrm{for\ } 0 \le x \le 1$

$\begin{matrix} E(X) &=& \frac{2}{3} \\ & & \\ \mathrm{Var}(X) &=& \frac{1}{18} \end{matrix}$

Distribution of two standard uniform variables

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

$f(x)=\left\{\begin{matrix} 4x & \mathrm{for\ }0 \le x < \frac{1}{2} \\ \\ 4-4x & \mathrm{for\ }\frac{1}{2} \le x \le 1 \end{matrix}\right.$

$F(x)=\left\{\begin{matrix} 2x^2 & \mathrm{for\ }0 \le x < \frac{1}{2} \\ \\ 1-2(1-x)^2 & \mathrm{for\ }\frac{1}{2} \le x \le 1 \end{matrix}\right.$

$\begin{matrix} E(X) &=& \frac{1}{2} \\ \\ \mathrm{Var}(X) &=& \frac{1}{24} \end{matrix}$

Distribution of the absolute difference of two standard uniform variables

This distribution for a=0, b=1 and c=0 is distribution of $X = | X 1 - X 2 |$ , where $X 1, X 2$ are two random variables with standard uniform distribution.

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

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

Use of the distribution

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" [1] as to the modal value.

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).

External links

Eric W. Weisstein, Triangular Distribution at MathWorld.
Triangle Distribution, decisionsciences.org
Triangular Distribution, brighton-webs.co.uk

v • d • e

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin-Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,8)

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-8,8)

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Category: Continuous distributions

Index Of Related Pages

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