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.

ParameterDescriptionSupport
`mu`Mean$-\infty <\mu <\infty$
`sigma`Scale parameter$\sigma \ge 0$

Probability Density Function

The probability density function (pdf) is

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