distributional (version 0.6.0)

dist_logistic: The Logistic distribution

Description

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.

Usage

dist_logistic(location, scale)

Arguments

location, scale

location and scale parameters.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_logistic.html

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):

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

Cumulative distribution function (c.d.f):

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

Moment generating function (m.g.f):

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

for \(-1 < st < 1\), where \(B(a, b)\) is the Beta function.

See Also

stats::Logistic

Examples

dist <- dist_logistic(location = c(5,9,9,6,2), scale = c(2,3,4,2,1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)