Logistic Distribution

Overview

The logistic distribution is used for growth models and in logistic regression. It has longer tails and a higher kurtosis than the normal distribution.

Parameters

The logistic distribution uses the following parameters.

Parameter	Description	Support
mu	Mean	$$-\infty <\mu <\infty$$
sigma	Scale parameter	$$\sigma \ge 0$$

Probability Density Function

The probability density function (pdf) is

$$f(x|\mu ,\sigma )=\frac{\exp \left\{\frac{x-\mu }{\sigma }\right\}}{\sigma {\left(1+\exp \left\{\frac{x-\mu }{\sigma }\right\}\right)}^2};-\infty <x<\infty .$$

Relationship to Other Distributions

The loglogistic distribution is closely related to the logistic distribution. If x is distributed loglogistically with parameters μ and σ, then log(x) is distributed logistically with parameters μ and σ.