# Logistic

##### 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.

##### 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

`library(stats)`

```
# NOT RUN {
var(rlogis(4000, 0, scale = 5)) # approximately (+/- 3)
pi^2/3 * 5^2
# }
```

*Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0*