Kurtosis

Definition

Let $X$ be a real random variable with $E[X^4] < \infty$ , mean $\mu$ and standard deviation $\sigma > 0$ . The kurtosis of $X$ is $\operatorname{Kurt}(X) = E\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{\mu_4}{\sigma^4},$ where $\mu_4$ is the fourth central moment. The excess kurtosis is $\operatorname{Kurt}(X) - 3$ .

Remarks

Every normal distribution has kurtosis $3$ , so its excess kurtosis is $0$ . For every $X$ , $\operatorname{Kurt}(X) \ge 1$ . The Bernoulli distribution with $p = 1/2$ reaches this bound. Distributions with positive excess kurtosis are called leptokurtic. Those with negative excess kurtosis are called platykurtic. The kurtosis does not change under $X \mapsto aX + b$ with $a \ne 0$ .

Tag	`0CBQVVNBMP`
URI	`m2://0CBQVVNBMP`
Kind	definition
Writing	W1 stated
MSC	62E10
Defines	Kurtosis
Uses	Central moment, Standard deviation, Normal distribution

Tag

0CBQVVNBMP

URI

m2://0CBQVVNBMP

Kind

definition

Writing

W1 stated

MSC

62E10

Defines

Kurtosis

Uses

Central moment, Standard deviation, Normal distribution

Kurtosis

Definition

Remarks

Reading path

Further reading

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