Editing Logistic distribution (section)

== Specification ==

=== Cumulative distribution function ===
The logistic distribution receives its name from its [[cumulative distribution function]], which is an instance of the family of logistic functions. The cumulative distribution function of the logistic distribution is also a scaled version of the [[Hyperbolic function|hyperbolic tangent]].

:<math>F(x; \mu, s) = \frac{1}{1+e^{-(x-\mu)/s}} = \frac12 + \frac12 \operatorname{tanh} \left(\frac{x-\mu}{2s}\right).</math>

In this equation {{math|''μ''}} is the [[mean]], and {{math|''s''}} is a scale parameter proportional to the [[standard deviation]].

=== Probability density function ===

The [[probability density function]] is the [[partial derivative]] of the cumulative distribution function:

: <math>
\begin{align}
f(x; \mu,s)  & = \frac{\partial F(x; \mu, s)}{\partial x} = \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} \\[4pt]
& =\frac{1}{s\left(e^{(x-\mu)/(2s)}+e^{-(x-\mu)/(2s)}\right)^2} \\[4pt]
& =\frac{1}{4s} \operatorname{sech}^2\left(\frac{x-\mu}{2s}\right).
\end{align}
</math>

When the location parameter&nbsp;{{math|''&mu;''}} is 0 and the scale parameter&nbsp;{{math|''s''}} is&nbsp;1, then the [[probability density function]] of the logistic distribution is given by

: <math>
\begin{align}
f(x; 0,1) & = \frac{e^{-x}}{(1+e^{-x})^2} \\[4pt]
& = \frac 1 {(e^{x/2} + e^{-x/2})^2} \\[5pt]
& = \frac 1 4 \operatorname{sech}^2 \left(\frac x 2 \right).
\end{align}
</math>

Because this function can be expressed in terms of the square of the [[hyperbolic function|hyperbolic secant function]] "sech", it is sometimes referred to as the ''sech-square(d) distribution''.<ref>Johnson, Kotz & Balakrishnan (1995, p.116).</ref> (See also: [[hyperbolic secant distribution]]).

=== Quantile function ===
The [[inverse function|inverse]] cumulative distribution function ([[quantile function]]) of the logistic distribution is a generalization of the [[logit]] function.  Its derivative is called the quantile density function.  They are defined as follows:

:<math>Q(p;\mu,s) = \mu + s \ln\left(\frac{p}{1-p}\right).</math>

:<math>Q'(p;s) = \frac{s}{p(1-p)}.</math>

=== Alternative parameterization ===
An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, <math>s</math>, in terms of the standard deviation, <math>\sigma</math>, using the substitution <math>s\,=\,q\,\sigma</math>, where <math>q\,=\,\sqrt{3}/{\pi}\,=\,0.551328895\ldots</math>.  The alternative forms of the above functions are reasonably straightforward.