# Logistic

0th

Percentile

##### The Logistic Distribution

Density, distribution function, quantile function and random generation for the logistic distribution with parameters location and scale.

Keywords
distribution
##### Usage
dlogis(x, location = 0, scale = 1, log = FALSE)
plogis(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qlogis(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rlogis(n, location = 0, scale = 1)
##### Arguments
x, q
vector of quantiles.
p
vector of probabilities.
n
number of observations. If length(n) > 1, the length is taken to be the number required.
location, scale
location and scale parameters.
log, log.p
logical; if TRUE, probabilities p are given as log(p).
lower.tail
logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
##### Details

If location or scale are omitted, they assume the default values of 0 and 1 respectively.

The Logistic distribution with location $= \mu$ and scale $= \sigma$ has distribution function $$F(x) = \frac{1}{1 + e^{-(x-\mu)/\sigma}}$$ and density $$f(x)= \frac{1}{\sigma}\frac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}$$

It is a long-tailed distribution with mean $\mu$ and variance $\pi^2/3 \sigma^2$.

##### Value

• dlogis gives the density, plogis gives the distribution function, qlogis gives the quantile function, and rlogis generates random deviates. The length of the result is determined by n for rlogis, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

##### Note

qlogis(p) is the same as the well known logit function, $logit(p) = \log p/(1-p)$, and plogis(x) has consequently been called the inverse logit.

The distribution function is a rescaled hyperbolic tangent, plogis(x) == (1+ tanh(x/2))/2, and it is called a sigmoid function in contexts such as neural networks.

• logit
• sigmoid

##### source

[dpq]logis are calculated directly from the definitions.

rlogis uses inversion.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapter 23. Wiley, New York.

Distributions for other standard distributions.

• Logistic
• dlogis
• plogis
• qlogis
• rlogis
##### Examples
library(stats) var(rlogis(4000, 0, scale = 5)) # approximately (+/- 3) pi^2/3 * 5^2
Documentation reproduced from package stats, version 3.3, License: Part of R 3.3

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