Triangular distribution

Triangular
Probability density function
Cumulative distribution function
Parameters	$a:~a\in (-\infty ,\infty )$ $b:~a<b\,$ $c:~a\leq c\leq b\,$
Support	$a\leq x\leq b\!$
PDF	$\begin{cases} 0 & \text{for } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x < c, \\[4pt] \frac{2}{b-a} & \text{for } x = c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c < x \le b, \\[4pt] 0 & \text{for } b < x. \end{cases}$
CDF	$\begin{cases} 0 & \text{for } x \leq a, \\[2pt] \frac{(x-a)^2}{(b-a)(c-a)} & \text{for } a < x \leq c, \\[4pt] 1-\frac{(b-x)^2}{(b-a)(b-c)} & \text{for } c < x < b, \\[4pt] 1 & \text{for } b \leq x. \end{cases}$
Mean	${\frac {a+b+c}{3}}$
Median	${\begin{cases}a+{\sqrt {\frac {(b-a)(c-a)}{2}}}&{\text{for }}c\geq {\frac {a+b}{2}},\\[6pt]b-{\sqrt {\frac {(b-a)(b-c)}{2}}}&{\text{for }}c\leq {\frac {a+b}{2}}.\end{cases}}$
Mode	$c\,$
Variance	${\frac {a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}}$
Skewness	${\frac {{\sqrt 2}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{{\frac {3}{2}}}}}$
Ex. kurtosis	$-{\frac {3}{5}}$
Entropy	${\frac {1}{2}}+\ln \left({\frac {b-a}{2}}\right)$
MGF	$2{\frac {(b\!-\!c)e^{{at}}\!-\!(b\!-\!a)e^{{ct}}\!+\!(c\!-\!a)e^{{bt}}}{(b-a)(c-a)(b-c)t^{2}}}$
CF	$-2{\frac {(b\!-\!c)e^{{iat}}\!-\!(b\!-\!a)e^{{ict}}\!+\!(c\!-\!a)e^{{ibt}}}{(b-a)(c-a)(b-c)t^{2}}}$

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

Special cases

Mode at a bound

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

\left.{\begin{matrix}f(x)&=&2x\\[8pt]F(x)&=&x^{2}\end{matrix}}\right\}{\text{ for }}0\leq x\leq 1

{\begin{aligned}E(X)&={\frac {2}{3}}\\[8pt]{\mathrm {Var}}(X)&={\frac {1}{18}}\end{aligned}}

Symmetric triangular distribution

This distribution for a = 0, b = 1 and c = 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, i.e., the distribution of X = (X₁ + X₂)/2, where X₁, X₂ are two independent random variables with standard uniform distribution in [0, 1].

f(x)={\begin{cases}4x&{\text{for }}0\leq x<{\frac {1}{2}}\\4-4x&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}

F(x)={\begin{cases}2x^{2}&{\text{for }}0\leq x<{\frac {1}{2}}\\2x^{2}-(2x-1)^{2}&{\text{for }}{\frac {1}{2}}\leq x\leq 1\end{cases}}

{\begin{aligned}E(X)&={\frac {1}{2}}\\[6pt]\operatorname {Var}(X)&={\frac {1}{24}}\end{aligned}}

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.

{\begin{aligned}f(x)&=2-2x{\text{ for }}0\leq x<1\\[6pt]F(x)&=2x-x^{2}{\text{ for }}0\leq x<1\\[6pt]E(X)&={\frac {1}{3}}\\[6pt]\operatorname {Var}(X)&={\frac {1}{18}}\end{aligned}}

Generating Triangular-distributed random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

{\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)\leq U<1\end{cases}}\end{matrix}}

^[1]

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

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).

References

External links

Weisstein, Eric W. "Triangular Distribution". MathWorld.

Triangle Distribution, decisionsciences.org
Triangular Distribution, brighton-webs.co.uk

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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