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Chapter 3: Problem 18

Compute the measures of skewness and kurtosis of a distribution which is \(N\left(\mu, \sigma^{2}\right) .\) See Exercises \(1.9 .14\) and \(1.9 .15\) for the definitions of skewness and kurtosis, respectively.

Short Answer

Expert verified
The skewness of the given normal distribution is 0 and its kurtosis is 3. If one considers excess kurtosis, it is 0.

Step by step solution

01

Recognizing the Nature of Distribution

First thing to note is that the normal distribution, represented by \(N(\mu, \sigma^{2})\), is a symmetric bell-shaped distribution. This has implications for its skewness and kurtosis.
02

Computing Skewness

The skewness of a normal distribution is 0. This is due to its symmetrical nature: the left and right tails of the distribution are exact mirror images of each other. Hence, the skewness for this distribution is 0.
03

Computing Kurtosis

The kurtosis of a normal distribution is 3. This results from the specific mathematical formula for kurtosis and the specific shape of the normal distribution. However, often we use excess kurtosis which is kurtosis - 3. So the excess kurtosis for this distribution is 3-3=0.

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