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Statistics[Distributions]

 Triangular
 triangular distribution

 Calling Sequence Triangular(a, b, c) TriangularDistribution(a, b, c)

Parameters

 a - lower bound b - upper bound c - distribution mode

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t

 subject to the following conditions:

$a\le c,c\le b,a

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

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Triangular}\left(a,b,c\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\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\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\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\$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{a}}{{3}}{+}\frac{{b}}{{3}}{+}\frac{{c}}{{3}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{1}}{{18}}{}{{a}}^{{2}}{+}\frac{{1}}{{18}}{}{{b}}^{{2}}{+}\frac{{1}}{{18}}{}{{c}}^{{2}}{-}\frac{{1}}{{18}}{}{a}{}{b}{-}\frac{{1}}{{18}}{}{a}{}{c}{-}\frac{{1}}{{18}}{}{b}{}{c}$ (4)

References

 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.