Triangular Distribution
Triangular Distribution#
The triangular distribution is a three-parameter continuous probability distribution. The table below summarizes some important aspects of the distribution.
Notation |
\(X \sim \mathcal{T}_r (a, b, c)\) |
Parameters |
\(a \in (-\infty, b)\) (lower bound) |
\(b \in (a, \infty)\) (upper bound) |
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\(c \in (a, b)\) (mid point) |
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\(\mathcal{D}_X = [a, b] \subset \mathbb{R}\) |
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\(f_X (x; a, b, c) = \begin{cases} 0.0 & x < a \\ \frac{2}{(b-a) (c-a)} (x - a) & x \in [a, c) \\ \frac{2}{b-a} & x = c \\ \frac{2}{(b-a) (b - c)} (b - x) & x \in (c, b] \\ 0.0 & x > b \end{cases}\) |
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\(F_X (x; a, b, c) = \begin{cases} 0.0 & x < a \\ \frac{(x - a)^2}{(b - a) (c - a)} & x \in [a, c] \\ 1.0 - \frac{(b - x)^2}{(b-a) (b - c)} & x \in (c, b] \\ 1.0 & x > b \end{cases}\) |
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\(F^{-1}_X (x; a, b, c) = \begin{cases} a + \sqrt{(b - a) (c - a) x} & x \in [0.0, \frac{c - a}{b - a}] \\ b - \sqrt{(b - a) (b - c) (1 - x)} & x \in (\frac{c - a}{b - a}, 1.0] \\ \end{cases}\) |
The plots of probability density functions (PDFs), sample histogram (of \(5'000\) points), cumulative distribution functions (CDFs), and inverse cumulative distribution functions (ICDFs) for different parameter values are shown below.
