Logistic distribution

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.

Logistic distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu ,}$ location (real) ${\displaystyle s>0,}$ scale (real)
Support	${\displaystyle x\in (-\infty ,\infty )}$
PDF	${\displaystyle {\frac {e^{-(x-\mu )/s}}{s\left(1+e^{-(x-\mu )/s}\right)^{2}}}}$
CDF	${\displaystyle {\frac {1}{1+e^{-(x-\mu )/s}}}}$
Quantile	${\displaystyle \mu +s\log \left({\frac {p}{1-p}}\right)}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle {\frac {s^{2}\pi ^{2}}{3}}}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle 6/5}$
Entropy	${\displaystyle \ln s+2}$
MGF	${\displaystyle e^{\mu t}\mathrm {B} (1-st,1+st)}$ for ${\displaystyle t\in (-1/s,1/s)}$ and ${\displaystyle \mathrm {B} }$ is the Beta function
CF	${\displaystyle e^{it\mu }{\frac {\pi st}{\sinh(\pi st)}}}$
Expected shortfall	${\displaystyle \mu +{\frac {sH(p)}{1-p}}}$ where ${\displaystyle H(p)}$ is the binary entropy function ${\displaystyle H(p)=-p\ln(p)-(1-p)\ln(1-p)}$