Logistic {stats} R Documentation

## 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].

### 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.

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