Triangular distribution

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

$f(x)=\left\{{\begin{matrix}{\frac {2(x-a)}{(b-a)(c-a)}}&\mathrm {for\ } a\leq x\leq c\\{\frac {2(b-x)}{(b-a)(b-c)}}&\mathrm {for\ } c<x\leq b\end{matrix}}\right.$

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

The cumulative distribution function is:

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

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

${\begin{matrix}E(X)={\frac {a+b+c}{3}}\\\\\mathrm {Var} (X)={\frac {a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}}\end{matrix}}$