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.

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]  3.724355  7.432998  6.580928  4.839895  7.347398  9.518203 -1.524048
#>  [8]  5.274608  9.105718  2.953511
#> 
#> [[2]]
#>  [1]  0.6391895 12.1972645 14.5751780  9.0928444 18.3878195 12.4067875
#>  [7]  8.4443315 20.3452265  8.6495240  8.7271871
#> 
#> [[3]]
#>  [1] 10.936644 10.538630  9.936112  8.860270 19.591185  3.530419  9.476767
#>  [8] 18.748411 11.604186  4.269333
#> 
#> [[4]]
#>  [1]  6.421523 10.586980  8.613862  3.333075  7.154727 12.556554  8.337592
#>  [8]  8.238406  6.382433  6.285357
#> 
#> [[5]]
#>  [1]  0.8136582  2.5203872  1.5520066  2.1138336  1.0700558  2.3728112
#>  [7]  2.8408122  5.7271590 -0.1118496  0.4188451
#> 

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

Arguments

Details

See also

Examples