Finally, note that the probability function in this case is modestly close to a bell shaped curve so we would expect a
kurtosis 4 and using
the alternate formulation 4 in the vicinity of 3. Indeed, noting that (conveniently)
\(\mu = 0\) gives
\begin{align*}
\text{Numerator = } & E[X^4] - 4 \mu E[X^3] + 6 \mu^2 E[X^2] - 3 \mu^4\\
& = \int_{-1}^1 x^4 \cdot \frac{3}{4} \cdot (1-x^2) dx\\
& = \frac{3}{4} \cdot (x^5 /5-x^7 /7) \big |_{-1}^1\\
& = \frac{3}{4} \cdot 2(1/5-1/7)\\
& = \frac{3}{35}
\end{align*}
and so by dividing by \(\sigma^4 = \sqrt{\frac{1}{5}}^4 = \frac{1}{25}\) gives a kurtosis of
\begin{equation*}
\gamma_2 = \frac{3}{35} / \frac{1}{25} = \frac{75}{35} \approx 2.14.
\end{equation*}