ABSTRACT
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.
