Triangular distribution

Triangular
Triangular
	Probability density function
	Cumulative distribution function
Parameters	; ;
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
MGF
CF

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.

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	${\begin{cases}0&{\text{for }}x<a,\\{\frac {2(x-a)}{(b-a)(c-a)}}&{\text{for }}a\leq x<c,\\[4pt]{\frac {2}{b-a}}&{\text{for }}x=c,\\[4pt]{\frac {2(b-x)}{(b-a)(b-c)}}&{\text{for }}c<x\leq b,\\[4pt]0&{\text{for }}b<x.\end{cases}}$
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	${\frac {{\sqrt {2}}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{\frac {3}{2}}}}$
Excess 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	$-2{\frac {(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}}{(b-a)(c-a)(b-c)t^{2}}}$