Triangular distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ triangular(a,m,b) is used to indicate that the random variable X has the triangular distribution with parameters a, m and b. A triangular random variable X has probability density function

f(x) = \{ \begin{matrix}\frac{2(x−a)}{(b−a)(m−a)} & a < x < m \\ \frac{2(b−x)}{(b−a)(b−m)} & m ≤ x < b. \end{matrix}

The triangular distribution can be used as an approximate model when there are no data values. An expert familiar with the population specifies a minium value a, a most likely value m, and a maximum value b. The probability density function is illustrated below.

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f(x)
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The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \{ \begin{matrix}\frac{(x−a)^{2}}{(b−a)(m−a)} & a < x < m \\ 1 − \frac{(b−x)^{2}}{(b−a)(b−m)} & m ≤ x < b. \end{matrix}

The survivor function on the support of X is

S(x) = P(X ≥ x) = \{ \begin{matrix}1 − \frac{(x−a)^{2}}{(b−a)(m−a)} & a < x < m \\ \frac{(b−x)^{2}}{(b−a)(b−m)} & m ≤ x < b. \end{matrix}

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \{ \begin{matrix}\frac{2(a−x)}{ab−mb+ma+x^{2}−2ax} & a < x < m \\ \frac{2}{b−x} & m ≤ x < b. \end{matrix}

The moment generating function of X is

M(t) = E [e^{tX}] = −\frac{2(−me^{ta}+ be^{ta}+ be^{tm}+ ae^{tm}− ae^{tb}+ me^{tb})}{(a − m)(a − b)(m − b)t^{2}} t > 0.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = \frac{2(−me^{ita}+ be^{ita}+ be^{itm}+ ae^{itm}− ae^{itb}+ me^{itb})}{(a − m)(a − b)(m − b)t^{2}} t > 0.

The population mean, variance, skewness, and kurtosis of X are

E[X] = \frac{a + m + b}{3} V [X] = \frac{a^{2}+ m^{2}+ b^{2}− ab − am − mb}{18}

E [( \frac{X − μ}{σ} ) ^{3}] = \frac{\sqrt{2}(a + b − 2m)(2a − b − m)(a − 2b + m)}{5(a^{2}+ b^{2}+ m^{2}− ab − am − bm)^{3/2}} E [( \frac{X − μ}{σ} ) ^{4}] = \frac{12}{5}.

APPL verification: The APPL statements

X := TriangularRV(a, m, b); CDF(X); SF(X); HF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the cumulative distribution function, survivor function, hazard function, population mean, variance, skewness, kurtosis, and moment generating function.