Kurtosis in terms of Non-Central Moments

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Theorem

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

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

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


Proof

\(\ds \alpha_4\) \(=\) \(\ds \expect {\paren {\dfrac {X - \mu} \sigma}^4}\) Definition of Kurtosis
\(\ds \) \(=\) \(\ds \dfrac {\expect {X^4 - 4 X^3 \mu + 6 X^2 \mu^2 - 4 X \mu^3 + \mu^4} } {\sigma^4}\) Fourth Power of Difference
\(\ds \) \(=\) \(\ds \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 4 \mu^3 \expect X + \mu^4} {\sigma^4}\) Linearity of Expectation Function
\(\ds \) \(=\) \(\ds \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4}\) $\mu = \expect X$

$\blacksquare$