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Concept# Logistic distribution

Summary

In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution.
Specification
Probability density function
When the location parameter μ is 0 and the scale parameter s is 1, then the probability density function of the logistic distribution is given by
:
\begin{align}
f(x; 0,1) & = \frac{e^{-x}}{(1+e^{-x})^2} \[4pt]
& = \frac 1 {(e^{x/2} + e^{-x/2})^2} \[5pt]
& = \frac 1 4 \operatorname{sech}^2 \left(\frac x 2 \right).
\end{align}
Thus in general the density is:
:
\begin{align}
f(x; \mu,s) & = \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} \[4pt]
& =\frac{1}{s\left(e^{

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