Triangular distribution - Definition and Overview

In probability theory and statistics, the triangular distribution is a continuous probability distribution with the probability density function defined on the interval [a, b]:

<math> f(x)=\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 < x \le b \end{matrix}\right. <math>

where a (location), b (scale) and c (shape) are the triangular distribution parameters.

The cumulative distribution function is:

<math> F(x)=\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 < x \le b \end{matrix}\right. <math>

The expected value and variance of a triangular random variable X are:

<math>

\begin{matrix}

 E(X) = \frac{a+b+c}{3} \\

\\

 \mathrm{Var}(X) = \frac{a^2+b^2+c^2-ab-ac-bc}{18}

\end{matrix} <math>