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Logistic distribution

What it describes

The Logistic distribution is symmetric with somewhat heavier tails than normal and a closed-form sigmoid CDF. It is the continuous law behind many growth curves and logistic models.

It spans the real line. mu is the symmetry point and sigma sets scale; its hazard has a bounded, non-constant form.

History and names

Pierre-François Verhulst introduced the logistic curve in 1838 for bounded population growth. The logistic distribution emerged later by reading that curve as a CDF.

How it connects to other distributions

It is the basis of logistic regression, where the CDF maps a linear predictor to probability. It resembles Normal and Laplace but has elementary quantiles and intermediate tails.

Where it appears

  • population growth and saturation curves
  • logistic regression and symmetric errors with moderate tails

Fitting cautions

Do not choose logistic just because a plot looks S-shaped: scale and tails must match the data-generating process.

From population growth to log odds

Verhulst introduced a sigmoid curve for limited population growth long before Logistic became a familiar probability distribution. Differentiating that cumulative curve produces the symmetric density. Another construction is revealing: the logit of a Uniform variable has a standard Logistic distribution.

This identity connects the law with odds and helps explain logistic regression, although the regression model specifies a conditional probability and does not require a continuous Logistic response. Compared with the Normal, Logistic places somewhat more mass in its tails and has an elementary CDF. That computational convenience should be paired with a comparison of quantiles, not merely centres and variances.

Decision guide

A good candidate when: a symmetric real-line distribution with tails somewhat heavier than Normal and a simple sigmoid CDF is needed.

Compare it with: Normal and Hyperbolic Secant. Match variances before comparing scales; Logistic scale is not a standard deviation.

References

  • SciPy reference: scipy.stats.logistic — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Verhulst, P.-F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121.
  • McCullagh, P. & Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman & Hall.

Logistic Distribution: equations and calculator

Distribution defintion

X∼Logistic(μ,σ)X\sim\mathrm{Logistic}\left(\mu,\sigma\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

μ∈R,σ∈R+\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=11+e−(x−μ)/σF_{X}\left(x\right)=\frac{1}{1+e^{-(x-\mu)/\sigma}}

Probability density function

fX(x)=e−(x−μ)/σσ(1+e−(x−μ)/σ)2f_{X}\left(x\right)=\frac{e^{-(x-\mu)/\sigma}} {\sigma\left(1+e^{-(x-\mu)/\sigma}\right)^2}

Percent point function/Sample

FX−1(u)=μ+σlog⁡(u1−u)F^{-1}_{X}\left(u\right)=\mu+\sigma \log\left(\frac{u}{1-u}\right)

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=μ\mathrm{Mean}(X)=\mu'_{1}=\mu

Parametric variance

Variance(X)=μ2′−μ1′2=σ2π23\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\sigma^2 \pi^2}{3}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+6/5\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3+6/5

Parametric median

Median(X)=μ\mathrm{Median}(X)=\mu

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}