chapter  22
10 Pages

- Logistic Distribution

The probability density function (pdf) of a logistic distribution with the location parameter a and scale parameter b is given by

f(x|a, b) = 1 b

exp {− ( x−a

)}[ 1 + exp

{− (x−a b

)}]2 , −∞ < x <∞, −∞ < a <∞, b > 0. (22.1)

The cumulative distribution function (cdf) is given by

F (x|a, b) = [ 1 + exp

{ − (x− a

b

)}]−1 . (22.2)

For 0 < p < 1, the inverse distribution function is

F−1(p|a, b) = a+ b ln[p/(1− p)]. (22.3)

The cdf and the inverse distribution function are in simple form, so they easy to calculate. The logistic distribution is symmetric about the location parameter a (see Figure 22.1), and it can be used as a substitute for a normal distribution.