The *chi‐distribution* is described by the PDF

The parameter *f* is often referred to as the *degrees of freedom*, in which case it should be restricted to integers. Figure 30.1 shows the effect of changing *f* with *α* fixed at 0 and *β* at 1.

With *f* set to 1, Equation (30.1) becomes almost the same as Equation (5.35), describing the normal distribution.

(30.2)

The only difference is a factor of 2 – required because it only applies to half the values of *x*, i.e. those greater than the mean. It is therefore known as the *half‐normal distribution*. Process disturbances and inferential errors, which are likely to have a mean close to zero, might follow a normal distribution. Their absolute value will then follow the half‐normal distribution.

Fitting to the absolute values of the NHV disturbances gives *α* as −0.08 and *β* as 1.31. As expected, because of the high kurtosis of the data, the fit is poor with *RSS* at 0.3248.

Mean and variance are

(30.3)

(30.4)

These give *μ* as 0.96 and *σ* as 0.52 – somewhat away from the ...

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