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.
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.
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.
Examples
set.seed(27)
X <- Logistic(2, 4)
X
#> [1] "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