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)

## Arguments

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

## Value

A Logistic object.

## Details

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

## Examples


set.seed(27)

X <- Logistic(2, 4)
X#> Logistic distribution (location = 2, scale = 4)
random(X, 10)#>  [1]  16.1520541  -7.5694209   9.7424712  -0.8466541  -3.0098187   0.4055911
#>  [7]  -8.1957130 -22.0364748  -5.3585558  -3.7506119
pdf(X, 2)#> [1] 0.0625log_pdf(X, 2)#> [1] -2.772589
cdf(X, 4)#> [1] 0.6224593quantile(X, 0.7)#> [1] 5.389191