Kurtosis in terms of Non-Central Moments

Theorem
Let $X$ be a random variable with expectation $\mu$ and standard deviation $\sigma$.

Then the kurtosis $\alpha_4$ of $X$ is given by:


 * $\ds \alpha_4 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4}$