Logistic distribution
The logistic distribution is a continuous probability distribution in probability theory and statistics, characterized by a location parameter \mu \in \mathbb{R} and a scale parameter s > 0, with probability density function f(x; \mu, s) = \frac{1}{s} \cdot \frac{e^{-(x - \mu)/s}}{(1 + e^{-(x - \mu)/s})^2} and cumulative distribution function F(x; \mu, s) = \frac{1}{1 + e^{-(x - \mu)/s}}.[1] This S-shaped CDF, known as the logistic function or sigmoid, arises naturally in models of bounded growth, and the distribution is symmetric about \mu, unimodal, and bell-shaped like the normal distribution but with heavier tails that allow for more extreme values.[1] For the standard case (\mu = 0, s = 1), the mean, median, and mode coincide at 0, with variance \pi^2 / 3 \approx 3.2899.[1] The logistic distribution traces its origins to the mid-19th century work of Belgian mathematician Pierre François Verhulst, who introduced the logistic growth model in 1838 to describe population dynamics with limited resources, where the CDF form emerged from the solution to the differential equation \frac{dP}{dt} = rP(1 - P/K).[2] This model was rediscovered independently in the early 20th century by Raymond Pearl and Lowell Reed in 1920 for U.S. population forecasting, leading to broader statistical adoption.[3] By the 1950s, David Cox formalized its use in logistic regression, a generalized linear model for binary outcomes where the log-odds follow a linear predictor, revolutionizing applications in biostatistics, economics, and machine learning.[3] Beyond regression, the distribution models phenomena with sigmoid growth patterns, such as in ecology for population limits and reliability engineering for failure times (e.g., mechanical component lifetimes).[4] Its mathematical tractability—closed-form CDF and moments—facilitates quantile estimation and simulation, making it a practical alternative to the normal distribution in scenarios requiring heavier tails without excessive complexity.[1] Variants like the generalized logistic extend it for skewness or multimodality in advanced modeling.[2]Definition
Probability density function
The probability density function of the logistic distribution is f(x \mid \mu, s) = \frac{\exp\left( \frac{x - \mu}{s} \right)}{s \left( 1 + \exp\left( \frac{x - \mu}{s} \right) \right)^2}, where \mu \in \mathbb{R} is the location parameter and s > 0 is the scale parameter.[5] This density is symmetric about \mu, producing a bell-shaped curve that peaks at x = \mu.[5] The logistic distribution exhibits heavier tails than the normal distribution with the same variance.[6] The variance is \frac{\pi^2 s^2}{3}.[5] In the standard parameterization (\mu = 0, s = 1), the random variable follows the distribution of the log-odds \log\left( \frac{p}{1-p} \right) for p drawn from a uniform distribution on [0, 1].[7] The PDF is derived by differentiating the cumulative distribution function of the logistic distribution.[5]Cumulative distribution function
The cumulative distribution function (CDF) of the logistic distribution with location parameter \mu and scale parameter s > 0 is given by F(x; \mu, s) = \frac{1}{1 + e^{-(x - \mu)/s}}, \quad x \in \mathbb{R}. [8][9] This CDF represents the standard logistic function, which produces an S-shaped curve that models cumulative probabilities for bounded outcomes ranging from 0 to 1, such as growth processes approaching saturation.[8] The function is strictly increasing over the real line, with F(\mu) = 1/2, and exhibits asymptotic behavior where F(x) \to 0 as x \to -\infty and F(x) \to [1](/page/1) as x \to \infty.[9] The logistic distribution derives its name from the logistic function's early role in 19th-century models of population growth, first introduced by Pierre François Verhulst in 1838 to describe sigmoid patterns in biological systems.[10]Properties
Moments and cumulants
The logistic distribution with location parameter \mu and scale parameter s > 0 has mean \mathbb{E}[X] = \mu and variance \mathrm{Var}(X) = s^2 \pi^2 / 3.[11] Due to the symmetry of the distribution about \mu, all odd central moments are zero, and thus the skewness is 0.[11] The kurtosis is $21/5 = 4.2, yielding an excess kurtosis of $6/5 = 1.2, which indicates that the logistic distribution is leptokurtic relative to the normal distribution (excess kurtosis 0).[11] The central moments m_n = \mathbb{E}[(X - \mu)^n] can be derived by direct integration of x^n against the probability density function or via the moment-generating function M(t) = \mathbb{E}[e^{tX}], which for the standard case (\mu = 0, s = 1) is M(t) = \Gamma(1 - it) \Gamma(1 + it), using the gamma function.[12] For the standard logistic random variable Z, the odd moments are m_n = 0 for odd n. For the general case, these scale as m_n = s^n m_n^{(Z)}, where m_n^{(Z)} denotes the corresponding moment of the standard distribution. For example, the second central moment is m_2 = \pi^2 / 3 and the fourth is m_4 = 7 \pi^4 / 15 for the standard case.[11] The cumulants \kappa_n are obtained as the coefficients in the Taylor expansion of the cumulant-generating function K(t) = \log M(t). The first cumulant is \kappa_1 = \mu, the second is \kappa_2 = s^2 \pi^2 / 3, the third is \kappa_3 = 0 (consistent with zero skewness), and higher cumulants follow from derivatives of K(t), which involve polygamma functions due to the gamma function representation of the moment-generating function.[12] For instance, the fourth cumulant is \kappa_4 = 2 s^4 \pi^4 / 15.[11] To highlight differences in shape, the first four central moments of the standard logistic distribution (mean 0, variance \pi^2 / 3 \approx 3.29) can be compared to those of the standard normal distribution (mean 0, variance 1). For fair scale comparison, consider the scaled logistic with variance 1 (achieved by dividing by \sqrt{\pi^2 / 3}):| Moment | Standard Normal | Scaled Logistic (Var=1) |
|---|---|---|
| Mean (\mu_1) | 0 | 0 |
| Variance (\mu_2) | 1 | 1 |
| Skewness (\mu_3 / \mu_2^{3/2}) | 0 | 0 |
| Excess Kurtosis (\mu_4 / \mu_2^2 - 3) | 0 | 1.2 |
Quantile function
The quantile function, also known as the inverse cumulative distribution function, of the logistic distribution with location parameter \mu and scale parameter s > 0 is given by Q(p; \mu, s) = \mu + s \ln\left(\frac{p}{1-p}\right), \quad 0 < p < 1. [12] This expression provides the value x such that the cumulative probability up to x equals p. The formula arises from the algebraic inversion of the cumulative distribution function. Setting F(x; \mu, s) = p and solving for x yields the logit transformation scaled and shifted by the parameters, as the logistic CDF is the inverse of this operation.[13] Key quantiles include the median Q(0.5; \mu, s) = \mu, reflecting the distribution's symmetry around the location parameter. The interquartile range, Q(0.75; \mu, s) - Q(0.25; \mu, s) = 2s \ln 3 \approx 2.197s, measures the spread of the central 50% of the distribution and scales linearly with s.[12] In simulation, the quantile function enables efficient generation of logistic random variables through inverse transform sampling: if U is a standard uniform random variable on (0,1), then X = \mu + s \ln\left(\frac{U}{1-U}\right) follows the logistic distribution with parameters \mu and s.[13] This method is particularly useful for Monte Carlo studies and bootstrapping in statistical modeling. The closed-form expression ensures numerical stability across p \in (0,1), with the logit-based computation avoiding overflow or cancellation issues common in other distributions, even for extreme quantiles near 0 or 1.[14] This property underpins its role in quantile regression, where it relates to the logit link function for modeling probabilities.Characteristic function
The characteristic function of a logistic random variable X with location parameter \mu and scale parameter s > 0 is \phi_X(t) = e^{i \mu t} \frac{\pi s t}{\sinh(\pi s t)}, where \sinh denotes the hyperbolic sine function.[15] An equivalent representation employs the gamma function: \phi_X(t) = e^{i \mu t} \Gamma(1 - i s t) \Gamma(1 + i s t). This equivalence follows from the identity |\Gamma(1 + i y)|^2 = \pi y / \sinh(\pi y) for real y, which is a consequence of the reflection formula for the gamma function.[15] To derive the characteristic function, compute the Fourier transform of the probability density function, \phi_X(t) = \mathbb{E}[e^{i t X}] = \int_{-\infty}^{\infty} e^{i t x} f_X(x) \, dx, where f_X(x) = \frac{e^{-(x-\mu)/s}}{s [1 + e^{-(x-\mu)/s}]^2}. One rigorous method avoids direct contour integration by expanding the standard logistic density (s=1, \mu=0) as an alternating infinite series of Laplace densities: f(x) = \sum_{k=1}^{\infty} (-1)^{k-1} \frac{k}{2} e^{-k |x|}. The characteristic function then emerges as the corresponding series summation, leveraging the known characteristic function of the Laplace distribution and identities from integral tables.[15] For the general case, apply the location-scale transformation to adjust for \mu and s. The hyperbolic form highlights key analytic properties: \phi_X(t) is defined and continuous for all real t, but its analytic continuation to the complex plane features simple poles along the imaginary axis at t = i k / s for nonzero integers k, arising from the zeros of \sinh(\pi s t). These poles, absent in the entire (pole-free) Gaussian characteristic function of the normal distribution, underscore the logistic's heavier tails and departure from Gaussianity.[15] Derivatives of the logarithm of the characteristic function at t=0 yield the cumulants via \kappa_n = \frac{1}{i^n} \frac{d^n}{dt^n} \log \phi_X(t) \big|_{t=0}. Using the gamma representation, \log \phi_X(t) = i \mu t + \log \Gamma(1 - i s t) + \log \Gamma(1 + i s t), the derivatives involve polygamma functions, as the digamma function \psi(z) = \frac{d}{dz} \log \Gamma(z) and higher-order polygamma functions \psi^{(n-1)}(z) = \frac{d^n}{dz^n} \log \Gamma(z) appear in the expansion.Parameterizations and estimation
Location-scale parameterization
The logistic distribution is commonly expressed in location-scale form using a location parameter μ ∈ ℝ, which serves as the mean and median, and a positive scale parameter s > 0, which controls the dispersion of the distribution.[11] The standard deviation of this parameterization is given by s π / √3, reflecting the fixed variance of π²/3 for the standard logistic distribution (with μ = 0 and s = 1) scaled by s².[11] This form facilitates modeling symmetric, bell-shaped data with heavier tails than the normal distribution, maintaining the characteristic S-shaped cumulative distribution function under shifts and stretches. An alternative rate parameterization replaces the scale s with a rate parameter β = 1/s > 0, which emphasizes the steepness of the distribution's rise and is particularly useful in contexts involving growth rates or odds ratios.[9] In this variant, the probability density function adjusts to f(x) = β e^{-β (x - μ)} / [1 + e^{-β (x - μ)}]^2, preserving the location μ while inverting the scale for interpretive convenience in rate-based models.[9] The logistic distribution belongs to the location-scale family, exhibiting shape invariance under affine transformations: if X follows a logistic distribution with parameters μ and s, then for any a ≠ 0 and b ∈ ℝ, the transformed variable Y = aX + b follows a logistic distribution with parameters aμ + b and |a|s.[9] This property ensures that the distribution's qualitative features, such as symmetry and kurtosis, remain unchanged after linear rescaling and shifting. Historically, the logistic form traces back to Pierre-François Verhulst's 1838 work on population dynamics, where he introduced a growth model incorporating a rate parameter to describe self-limiting exponential growth toward a carrying capacity, laying the foundation for the distribution's parameterization in ecological and demographic applications.[16] In Verhulst's original formulation, the growth rate r corresponds conceptually to the inverse scale (akin to β), highlighting the distribution's roots in bounded growth processes. Conversions between the scale and rate parameterizations are straightforward, as summarized below:| Scale Parameter (s) | Rate Parameter (β) |
|---|---|
| s > 0 | β = 1/s |
| Standard (s = 1) | β = 1 |
| Variance = s² π² / 3 | Variance = π² / (3 β²) |