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Statistics for Process Control Engineers by Myke King

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22Logistic Distribution

22.1 Logistic

A generalised form of the logistic distribution is described by

(22.1)images

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

(22.2)images
(22.3)images

The CDF is

The CDF can be inverted

(22.5)images

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, 0.25β). Fitting Equation (22.4) to the C4 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.

x vs. f(x) displaying a bell curve for δ = 0, a right skewed-curve for δ = 0.7, and a left-skewed curve for δ = 0.5, illustrating the effect of changing of changing δ with α fixed at 0 and ϐ at 1.

Figure 22.1 Logistic: Effect of δ on shape

The mean and variance are

(22.6)images
(22.7)images

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

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