A continuous distribution on the real line. For binary outcomes
the model given by \(P(Y = 1 | X) = F(X \beta)\) where
\(F\) is the Logistic `cdf()`

is called *logistic regression*.

Logistic(location = 0, scale = 1)

location | The location parameter for the distribution. For Logistic distributions, the location parameter is the mean, median and also mode. Defaults to zero. |
---|---|

scale | The scale parameter for the distribution. Defaults to one. |

A `Logistic`

object.

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

In the following, let \(X\) be a Logistic random variable with
`location`

= \(\mu\) and `scale`

= \(s\).

**Support**: \(R\), the set of all real numbers

**Mean**: \(\mu\)

**Variance**: \(s^2 \pi^2 / 3\)

**Probability density function (p.d.f)**:

$$ f(x) = \frac{e^{-(\frac{x - \mu}{s})}}{s [1 + \exp(-(\frac{x - \mu}{s})) ]^2} $$

**Cumulative distribution function (c.d.f)**:

$$ F(t) = \frac{1}{1 + e^{-(\frac{t - \mu}{s})}} $$

**Moment generating function (m.g.f)**:

$$ E(e^{tX}) = e^{\mu t} \beta(1 - st, 1 + st) $$

where \(\beta(x, y)\) is the Beta function.

Other continuous distributions: `Beta`

,
`Cauchy`

, `ChiSquare`

,
`Exponential`

, `FisherF`

,
`Gamma`

, `LogNormal`

,
`Normal`

, `StudentsT`

,
`Tukey`

, `Uniform`

,
`Weibull`

#> Logistic distribution (location = 2, scale = 4)#> [1] 16.1520541 -7.5694209 9.7424712 -0.8466541 -3.0098187 0.4055911 #> [7] -8.1957130 -22.0364748 -5.3585558 -3.7506119#> [1] 0.0625#> [1] -2.772589#> [1] 0.6224593#> [1] 5.389191