Excess Kurtosis of Gamma Distribution/Proof 2

Proof
From the definition of excess kurtosis, we have:


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

where:
 * $\mu = \expect X$ is the expectation of $X$
 * $\sigma = \sqrt {\var X}$ is the standard deviation of $X$.

By Expectation of Gamma Distribution, we have:


 * $\mu = \dfrac \alpha \beta$

By Variance of Gamma Distribution, we have:


 * $\sigma^2 = \dfrac \alpha {\beta^2}$

From Expectation of Power of Gamma Distribution‎, we have:


 * $\expect {X^n} = \dfrac {\alpha^{\overline n} } {\beta^n}$

where $\alpha^{\overline n}$ denotes the $n$th rising factorial of $\alpha$.

Hence: