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Triangular Distribution

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TriangularDistribution

The triangular distribution is a continuous distribution defined on the range x in [a,b] with probability density function

P(x)={(2(x-a))/((b-a)(c-a)) for a<=x<=c; (2(b-x))/((b-a)(b-c)) for c<x<=b
(1)

and distribution function

D(x)={((x-a)^2)/((b-a)(c-a)) for a<=x<=c; 1-((b-x)^2)/((b-a)(b-c)) for c<x<=b,
(2)

where c in [a,b] is the mode.

The symmetric triangular distribution on [a,b] is implemented in the Wolfram Language as TriangularDistribution[a, b], and the triangular distribution on [a,b] with mode c as TriangularDistribution[a, b, c].

The mean is

mu=1/3(a+b+c),
(3)

the raw moments are

mu_2^'=1/6(a^2+b^2+c^2+ab+ac+bc)
(4)
mu_3^'=1/(10)(a^3+b^3+c^3+a^2b+a^2c+b^2a+b^2c+c^2a+c^2b+abc),
(5)

and the central moments are

mu_2=1/(18)(a^2+b^2+c^2-ab-ac-bc)
(6)
mu_3=-1/(270)(a+b-2c)(a+c-2b)(b+c-2a)
(7)
mu_4=1/(135)(a^2+b^2+c^2-ab-ac-bc)^2.
(8)

It has skewness and kurtosis excess given by

gamma_1=(sqrt(2)(a+b-2c)(2a-b-c)(a-2b+c))/(5(a^2+b^2+c^2-ab-ac-bc)^(3/2))
(9)
gamma_2=-3/5.
(10)

See also

Triangle Function, Uniform Distribution

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References

Evans, M.; Hastings, N.; and Peacock, B. "Triangular Distribution." Ch. 40 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 187-188, 2000.

Referenced on Wolfram|Alpha

Triangular Distribution

Cite this as:

Weisstein, Eric W. "Triangular Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangularDistribution.html

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