Kurtosis

Kurtosis

If we let $\mu_4 = E\left[(X-\mu)^4 \right]$ i.e. the fourth central moment

Then the kurtosis is defined as $\frac{\mu_4}{\sigma^4}$ or $\frac{\mu_4}{\mu_2^2}$

where $\mu_2 = E\left[(X-\mu)^2 \right]$ and $\mu = E(X)$

The point of dividing by $\sigma^4$ is that we do not want general variance to appear like peakedness

Note that for $X \sim N(\mu, \sigma^2)$: $\frac{\mu_4}{\sigma_4}=3$

Therefore we define excess kurtosis as: $\frac{\mu_4}{\sigma_4} - 3$