Truncated Gumbel (max.) Distribution
Truncated Gumbel (max.) Distribution#
The truncated Gumbel (max.) distribution is a four-parameter continuous probability distribution. The table below summarizes some important aspects of the distribution.
Notation |
\(X \sim \mathrm{Gumbel}_{\mathrm{Tr}} (\mu, \beta, a, b)\) |
Parameters |
\(\mu \in \mathbb{R}\) (location parameter) |
\(\beta > 0\) (scale parameter) |
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\(a \in (-\infty, b)\) (lower bound) |
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\(b \in (a, \infty)\) (upper bound) |
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\(\mathcal{D}_X = [a, b]\) |
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\(f_X (x; \mu, \sigma, a, b) = \begin{cases} \frac{1}{F_{\mathrm{Gumbel}}(b; \mu, \sigma) - F_{\mathrm{Gumbel}}(a; \mu, \sigma)} f_{\mathrm{Gumbel}}(x; \mu, \sigma) & x \in [a, b]\\ 0.0 & x \notin [a, b] \end{cases}\) |
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\(F_X (x; \mu, \sigma, a, b) = \begin{cases} 0.0 & x < a \\ \frac{F_{\mathrm{Gumbel}}(x; \mu, \sigma) - F_{\mathrm{Gumbel}}(a; \mu, \sigma)}{F_{\mathrm{Gumbel}}(b; \mu, \sigma) - F_{\mathrm{Gumbel}}(a; \mu, \sigma)} & x \in [a, b] \\ 1.0 & x > b \end{cases}\) |
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\(F^{-1}_X (x; \mu, \beta, a, b) = \left(F_{\mathrm{Gumbel}}(b; \mu, \sigma) - F_{\mathrm{Gumbel}}(a; \mu, \sigma)\right) x + F_{\mathrm{Gumbel}}(a; \mu, \sigma)\) |
In the table above, \(f_{\mathrm{Gumbel}}\), \(F_{\mathrm{Gumbel}}\), and \(F^{-1}_{\mathrm{Gumbel}}\) are the probability density, the cumulative, and the inverse cumulative distribution functions of the (untruncated) Gumbel (max.) distribution, respectively.
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.
