A continuous distribution on the real line. For binary outcomes
the model given by P(Y = 1 | X) = F(X β) where
F is the Logistic
cdf() is called logistic regression.
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.
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 = μ and
scale = s.
Support: R, the set of all real numbers
Variance: s^2 π^2 / 3
Probability density function (p.d.f):
f(x) = e^(-(t - μ) / s) / (s (1 + e^(-(t - μ) / s))^2)
Cumulative distribution function (c.d.f):
F(t) = 1 / (1 + e^(-(t - μ) / s))
Moment generating function (m.g.f):
E(e^(tX)) = = e^(μ t) β(1 - st, 1 + st)
where β(x, y) is the Beta function.
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