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.

scale

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.

See also

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] 0.0625
log_pdf(X, 2)
#> [1] -2.772589
cdf(X, 4)
#> [1] 0.6224593
quantile(X, 0.7)
#> [1] 5.389191