20Normal Distribution

In Chapter 5 we covered the classic normal distribution along with its bivariate version. Chapter 12 addressed the lognormal distribution – one of the common variations. Here we cover the wide range of other variations, many of which better fit our examples.

20.1 Skew‐Normal

As its name suggests the skew‐normal distribution is a modified form of the normal distribution that can take account of skewness. It is described by the PDF

The parameter δ determines skewness. Figure 20.1 shows its effect, with α set to 0 and β to 1. If δ is less than zero the distribution is skewed to the left, if greater than zero it is skewed to the right. If set to −1 or +1 we obtain the half‐normal distribution that we will describe later.

f(x) versus x displaying 3 overlapping bell-shaped curves labeled δ = −0.9, δ = 0, and δ = 0.8 and 1 ascending, descending curve labeled δ = −1.

Figure 20.1 Skew‐normal: Effect of δ on shape

The mean, variance, skewness and kurtosis are

Figure 20.2 plots ...

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