stats (version 3.3)

Logistic: The Logistic Distribution

Description

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

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, $P[X > x]$.

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.

concept

• logit
• sigmoid

source

[dpq]logis are calculated directly from the definitions.

rlogis uses inversion.

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 $\pi^2/3 \sigma^2$.

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.

See Also

Distributions for other standard distributions.

Examples

var(rlogis(4000, 0, scale = 5))  # approximately (+/- 3)
pi^2/3 * 5^2