Problem: Given $X_1,X_2,\ldots,X_n$ are i.i.d with Ber($\theta$) distribution and let $\overline{X} = \dfrac{1}{n}\sum_{i=1}^{n} X_i$. Compute the Var$\left(\overline{X}(1-\overline{X})\right)$.
Here is my approach to this problem: $$\text{Var}(\left(\overline{X}(1-\overline{X})\right) = \mathbb{E}[\overline{X}^2(1-\overline{X})^2] - \mathbb{E}[\overline{X}(1-\overline{X})]^2.$$ Now to compute this, I have to consider the moment-generating function and calculate its derivative at zero to the fourth order. This work seems a little bit difficult about the transformation. I wonder that there is any way to solve this problem easier.