Editing Logistic distribution (section)

== Related distributions ==
* Logistic distribution mimics the [[sech distribution]]; they are different cases of the [[Champernowne distribution]].
* If <math>X \sim \mathrm{Logistic}(\mu, s)</math> then <math>kX + \ell \sim \mathrm{Logistic}(k\mu + \ell, |k|s)</math>.
* If <math>X \sim </math> [[Uniform distribution (continuous)|U(0, 1)]] then <math>\mu + s \cdot \text{logit}(X) \sim \mathrm{Logistic}(\mu, s)</math>, where <math>\text{logit}(X)=\log X-\log(1-X)</math> is the [[logit]] function.
* If <math>X \sim \mathrm{Gumbel}(\mu_X, \beta) </math> and <math> Y \sim \mathrm{Gumbel}(\mu_Y, \beta) </math> independently then <math> X-Y \sim \mathrm{Logistic}(\mu_X-\mu_Y,\beta) \,</math>.
* If <math>X </math> and <math>Y \sim \mathrm{Gumbel}(\mu, \beta) </math> then <math>X+Y \nsim \mathrm{Logistic}(2 \mu,\beta) \,</math> (The sum is ''not'' a logistic distribution). <math> E(X+Y) = 2\mu+2\beta\gamma \neq 2\mu = E\left(\mathrm{Logistic}(2 \mu,\beta) \right) </math>.
* If ''X'' ~ Logistic(''μ'', ''s'') then exp(''X'') ~ [[log-logistic distribution|LogLogistic]]<math> \left( \alpha = e^\mu, \beta = \frac 1 s \right) </math>,  and  exp(''X'') + ''γ'' ~ [[shifted log-logistic distribution|shifted log-logistic]]<math> \left( \alpha = e^\mu, \beta = \frac 1 s, \gamma \right) </math>.
* If ''X'' ~ [[Exponential distribution|Exponential(1)]] then 
::<math>\mu+s\log(e^X -1) \sim \operatorname{Logistic}(\mu,s). </math>
* If ''X'', ''Y'' ~ Exponential(λ) independently then
::<math>\mu+s\log\left(\frac X Y \right) \sim \operatorname{Logistic}(\mu,s).</math>
* The [[metalog distribution]] is generalization of the logistic distribution, in which power series expansions in terms of <math>p</math> are substituted for logistic parameters <math>\mu</math> and <math>\sigma</math>. The resulting metalog quantile function is highly shape flexible, has a simple closed form, and can be fit to data with linear least squares.