The Logistic distribution

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.

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
#> <distribution[5]>
#> [1] Logistic(5, 2) Logistic(9, 3) Logistic(9, 4) Logistic(6, 2) Logistic(2, 1)
mean(dist)
#> [1] 5 9 9 6 2
variance(dist)
#> [1] 13.159473 29.608813 52.637890 13.159473  3.289868
skewness(dist)
#> [1] 0 0 0 0 0
kurtosis(dist)
#> [1] 1.2 1.2 1.2 1.2 1.2

generate(dist, 10)
#> [[1]]
#>  [1]  3.8658619 10.3305153  6.3075558  5.1084827  5.5845942  7.4232965
#>  [7] -1.9331378  0.6473872  3.3505919  4.4773425
#> 
#> [[2]]
#>  [1]  8.242745  6.025529 15.206772 15.418833  9.202131 10.201776 10.916190
#>  [8] -0.232640 15.644106  4.768615
#> 
#> [[3]]
#>  [1] 14.7209662  8.0741974 13.0009167  6.9191037 -1.4026373  7.8970113
#>  [7]  9.8700286  0.5636868 17.8939036 -5.1062678
#> 
#> [[4]]
#>  [1]  1.570022  5.698264  2.746287  6.103908 11.612174  7.486487  5.364641
#>  [8] 10.789394  6.569626  4.660303
#> 
#> [[5]]
#>  [1]  3.5650681  3.1095549  0.8183330  1.3335869  1.3627772  1.1627231
#>  [7]  2.5062814  0.8852229 -0.2424980  0.5363091
#> 

density(dist, 2)
#> [1] 0.07457323 0.02686172 0.03153231 0.05249679 0.25000000
density(dist, 2, log = TRUE)
#> [1] -2.595974 -3.617053 -3.456743 -2.947003 -1.386294

cdf(dist, 4)
#> [1] 0.3775407 0.1588691 0.2227001 0.2689414 0.8807971

quantile(dist, 0.7)
#> [1]  6.694596 11.541894 12.389191  7.694596  2.847298