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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.

The probability of a positive delay is the integral of its density:Adding the separate no-delay event givesThis is the normalization of a probability distribution with both discrete and continuous parts.

A probability density function must be nonnegative. The printed condition $s\ne 0$ is therefore too weak: for $s<0$ the given prefactor makes the function negative everywhere. The location-scale form of the logistic distribution requires $s>0$, which is assumed in the remaining parts.

Set $t=(x-\mu )/(2s)$, so $dx=2sdt$. The normalization of a probability distribution follows from the antiderivative of the squared hyperbolic secant:For completeness, the same expression with $s<0$ integrates to $-1$, since the transformed limits reverse. If a signed scale were desired, replacing its prefactor by $1/(4|s|)$ would give a valid density, but that is a correction to the printed formula.