Logistic distribution

The logistic location family has density and distribution function

For a real location parameter $\mu$ and positive scale parameter $s$, the logistic distribution has probability density function and cumulative distribution functionDifferentiating $F$ 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.