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

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.

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.

See Also

Distributions for other standard distributions.

Examples

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