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.
Special cases
Mode at a bound The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the PDF and CDF become:
f ( x ) = 2 x , F ( x ) = x 2 {\displaystyle {\begin{aligned}f(x)&=2x,\\[8pt]F(x)&=x^{2}\end{aligned}}} for 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} .
E ( X ) = 2 3 Var ( X ) = 1 18 {\displaystyle {\begin{aligned}\operatorname {E} (X)&={\frac {2}{3}}\\[8pt]\operatorname {Var} (X)&={\frac {1}{18}}\end{aligned}}}
Distribution of the absolute difference of two standard uniform variables This distribution for a = 0, b = 1 and c = 0 is the distribution of X = |X1 − X2|, where X1, X2 are two independent random variables with standard uniform distribution.
f ( x ) = 2 − 2 x for 0 ≤ x < 1 F ( x ) = 2 x − x 2 for 0 ≤ x < 1 E ( X ) = 1 3 Var ( X ) = 1 18 {\displaystyle {\begin{aligned}f(x)&=2-2x{\text{ for }}0\leq x<1\\[6pt]F(x)&=2x-x^{2}{\text{ for }}0\leq x<1\\[6pt]E(X)&={\frac {1}{3}}\\[6pt]\operatorname {Var} (X)&={\frac {1}{18}}\end{aligned}}}
Symmetric triangular distribution The symmetric case arises when c = (a + b) / 2. In this case, an alternate form of the distribution function is:
f ( x ) = ( b − c ) − | c − x | ( b − c ) 2 = 2 b − a ( 1 − | a + b − 2 x | b − a ) {\displaystyle {\begin{aligned}f(x)&={\frac {(b-c)-|c-x|}{(b-c)^{2}}}\\[6pt]&={\frac {2}{b-a}}\left(1-{\frac {\left|a+b-2x\right|}{b-a}}\right)\end{aligned}}}
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