3 Gumbel and Exponential Distributions
3.1 The Gumbel Distribution
The Gumbel cumulative distribution function is:
The Gumbel CDF is continuous on \(\mathbb {R}\).
A random field \(Y : \text{Edge} \to \Omega \to \mathbb {R}\) is a Gumbel grid if:
The random variables \(\{ Y_e\} _{e \in \text{Edge}}\) are independent, and
For each edge \(e\) and \(x \in \mathbb {R}\), \(\mathbb {P}(Y_e \leq x) = F_{\text{Gumbel}}(x)\).
3.2 Transformation to Exponential Distribution
If \(Y\) is a Gumbel random variable, then for any \(y \in \mathbb {R}\):
This follows from the continuity of the Gumbel CDF. For any continuous CDF \(F\), the probability of a singleton is the difference \(F(y) - F(y^-)\), which is zero when \(F\) is continuous.
If \(Y\) is a Gumbel random variable, then:
If \(Y\) is a Gumbel random variable, then \(\exp (-Y)\) has the exponential distribution with rate 1. Specifically, for \(x \geq 0\):
For \(x {\gt} 0\), we have:
If \(Y\) is a Gumbel grid, then \(E_e = \exp (-Y_e)\) forms a grid of independent exponential random variables with rate 1.
The independence follows from the fact that \(\exp \) is a measurable function and composition with independent random variables preserves independence. The CDF property follows from the previous lemma applied to each edge.