The triangular distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

Note that the Triangular command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Triangular}\left(a\,b\,c\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<a\\ \frac{2\left(u-a\right)}{\left(b-a\right)\left(c-a\right)}& u\le c\\ \frac{2\left(b-u\right)}{\left(b-a\right)\left(-c+b\right)}& u\le b\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\left\{\begin{array}{cc}0.& 0.5<a\\ \frac{2.\left(0.5-1.a\right)}{\left(b-1.a\right)\left(c-1.a\right)}& 0.5\le c\\ \frac{2.\left(b-0.5\right)}{\left(b-1.a\right)\left(b-1.c\right)}& 0.5\le b\\ 0.& \mathrm{otherwise}\end{array}\right.$

$\mathrm{Mean}\left(X\right)$

$\frac{a}{3}+\frac{b}{3}+\frac{c}{3}$

$\mathrm{Variance}\left(X\right)$

$\frac{1}{18}{a}^2+\frac{1}{18}{b}^2+\frac{1}{18}{c}^2-\frac{1}{18}ab-\frac{1}{18}ac-\frac{1}{18}bc$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.