Logistic Distribution
X ~ Logistic(μ, s)
Symmetric / Growth-Curve Family • Published July 22, 2026
The distribution behind logistic regression's sigmoid: history from Verhulst's 1830s-40s population-growth curve to its role as the latent-variable error distribution that makes logistic regression's P(Y=1|x)=sigmoid(xβ) formula exact (confirmed by simulation), the only continuous CDF on this site needing no special function, constant excess kurtosis of 6/5 echoing the already-published Laplace, and a transcendental maximum-likelihood fit mirroring the iterative solving logistic regression itself requires.
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About This Reference Sheet
The distribution behind logistic regression's sigmoid: history from Verhulst's 1830s-40s population-growth curve to its role as the latent-variable error distribution that makes logistic regression's P(Y=1|x)=sigmoid(xβ) formula exact (confirmed by simulation), the only continuous CDF on this site needing no special function, constant excess kurtosis of 6/5 echoing the already-published Laplace, and a transcendental maximum-likelihood fit mirroring the iterative solving logistic regression itself requires.
Support
x ∈ (-∞, ∞)
Parameters
μ ∈ ℝ (location), s > 0 (scale)