Laplace Distribution
X ~ Laplace(μ, b)
Two-Sided Exponential / Heavy-Tailed Family • Published July 30, 2026
The 'two exponentials back to back' distribution: history from Pierre-Simon Laplace's 1774 work, a constant excess kurtosis of 3 regardless of scale (heavier-tailed than the Normal at every b), a median-based maximum-likelihood estimator that previews robust statistics, a direct tie to the Lasso's L1 penalty via its role as a Bayesian MAP prior, and simulation via the difference of two independent Exponentials.
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About This Reference Sheet
The 'two exponentials back to back' distribution: history from Pierre-Simon Laplace's 1774 work, a constant excess kurtosis of 3 regardless of scale (heavier-tailed than the Normal at every b), a median-based maximum-likelihood estimator that previews robust statistics, a direct tie to the Lasso's L1 penalty via its role as a Bayesian MAP prior, and simulation via the difference of two independent Exponentials.
Support
x ∈ (-∞, ∞)
Parameters
μ ∈ ℝ (location), b > 0 (scale)