A generalised form of the *logistic distribution* is described by

(22.1)

Depending on the sign of *δ*, the distribution is either lower‐bounded or upper‐bounded.

(22.2)

(22.3)

The CDF is

The CDF can be inverted

(22.5)

Figure 22.1 shows the effect of changing *δ* with *α* fixed at 0 and *β* at 1. Note that all the curves pass through the point (*α*, 0.25*β*). Fitting Equation (22.4) to the C_{4} in propane data gives *α* as 2.30, *β* as 1.27 and *δ* as 0.360. *RSS* is 0.0529, m\aking the distribution a relative poor choice.

The mean and variance are

(22.6)

(22.7)

As *δ* → 0, from Equation (11.4), we obtain the density functions ...

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