Excess Kurtosis of Exponential Distribution

Theorem
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$.

Then the excess kurtosis $\gamma_2$ of $X$ is equal to $6$.

Proof
From the definition of excess kurtosis, we have:


 * $\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$

where:
 * $\mu$ is the expectation of $X$.
 * $\sigma$ is the standard deviation of $X$.

By Expectation of Exponential Distribution we have:


 * $\mu = \beta$

By Variance of Exponential Distribution we have:


 * $\sigma = \beta$

So: