Equivalence of Definitions of Kurtosis

Theorem

The following definitions of the concept of Kurtosis are equivalent:

The kurtosis of $X$ is the fourth standardized moment of $X$:

$\alpha_4 = \expect {\paren {\dfrac {X - \mu} \sigma}^4}$

where $\expect {\, \cdot \,}$ denotes expectation.

The kurtosis of $X$ is defined as:

$\alpha_4 = \dfrac {\mu_4} {\paren {\mu_2}^2}$

where $\mu_i$ denotes the $i$th central moment of $X$.

$\ds \expect{\paren {\frac {X - \mu} \sigma}^4}$

$=$

$\ds \frac {\expect{\paren {X - \mu}^4} } {\sigma^4}$

$\ds $

$=$

$\ds \frac {\expect{\paren {X - \mu}^4} } {\paren {\var X}^2}$

$\ds $

$=$

$\ds \frac {\expect{\paren {X - \mu}^4} } {\paren {\expect{\paren {X - \expect X}^2} }^2}$

Definition of Variance

$\ds $

$=$

$\ds \frac {\expect{\paren {X - \mu}^4} } {\paren {\expect{\paren {X - \mu}^2} }^2}$

Definition of Expectation

$\ds $

$=$

$\ds \frac {\mu_4} {\paren {\mu_2}^2}$

Definition of Central Moment

$\blacksquare$