Logistic                package: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 <= 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' = m and 'scale' = s has
     distribution function

                    F(x) = 1 / (1 + exp(-(x-m)/s))

     and density

            f(x) = 1/s exp((x-m)/s) (1 + exp((x-m)/s))^-2.


     It is a long-tailed distribution with mean m and variance pi^2 /3
     s^2.

Value:

     'dlogis' gives the density, 'plogis' gives the distribution
     function, 'qlogis' gives the quantile function, and 'rlogis'
     generates random deviates.

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:

     '[dpr]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.

Examples:

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

