= Location-scale form of the logistic distribution
{title2=$f(x)=\operatorname{sech}^2((x-\mu)/(2s))/(4s),\quad s>0$}
For a real location parameter $\mu$ and positive scale parameter $s$, the <logistic distribution> has <probability density function> and <cumulative distribution function>
$$
f(x)=\frac1{4s}\operatorname{sech}^2\!\left(\frac{x-\mu}{2s}\right),\qquad
F(x)=\frac12\left[1+\tanh\!\left(\frac{x-\mu}{2s}\right)\right].
$$
Differentiating $F$ recovers $f$, and its limits at infinity show <normalization of a probability distribution>. The density is symmetric about $\mu$ with exponentially decaying tails, so its <expected value> exists and equals $\mu$. Allowing $s<0$ in the displayed expression makes $f$ negative and its integral $-1$; a signed scale must instead be replaced by $|s|$ in the prefactor to define the same valid density.
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