Logistic distribution

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CDF

1 1 + e − ( x − μ ) / s = 1 + tanh ⁡ x − μ 2 s 2 {\displaystyle {\frac {1}{1+e^{-(x-\mu )/s}}}={\frac {1+\tanh {\frac {x-\mu }{2s}}}{2}}}

CF

e i t μ π s t sinh ⁡ ( π s t ) {\displaystyle e^{it\mu }{\frac {\pi st}{\sinh(\pi st)}}}

Entropy

ln ⁡ s + 2 {\displaystyle \ln s+2}

Excess kurtosis

6 / 5 {\displaystyle 6/5}

Mean

μ {\displaystyle \mu }

Median

μ {\displaystyle \mu }