Example of using Delta Method Let $\hat p$ be the proportion of successes in $n$ independent Bernoulli trials each having probability $p$ of success.
(a) Compute the expectation of $\hat p (1-\hat p)$.
(b) Compute the approximate mean and variance of $\hat p (1-\hat p)$ using the Delta Method.
For part (a), I can calculate the expectation of $\hat p$ but got stuck on the expectation of $\hat p^2$, though I doubt this is a right track.
For part (b), I'm new to Delta method and as for the approximate variance I know I need to calculate the derivative of the function $\hat p (1-\hat p)$, but not quite sure. I appreciate if anyone can provide some answers or some similar examples online.

Let $$ k = #{success in n independent Bernoulli trials}$$ then we have $$ k \sim B(n,p)$$ and $$ E(\hat{p})=E(\frac{k}{n})=\frac{E(k)}{n}=\frac{np}{n}=p  $$ $$Var(\hat{p})=Var(\frac{k}{n})=\frac{Var(k)}{n^{2}}=\frac{np(1-p)}{n^{2}}=\frac{p(1-p)}{n}$$ thus $$E(\hat{p}^{2})=Var(\hat{p})+ E(\hat{p})^{2}=\frac{p-(1-n)p^{2}}{n} $$ this answers your first question.$$$$
For the second part,$$
Var(\hat{p}(1-\hat{p}))\approx(1-2\hat{p})^{2}Var(\hat{p})=(1-2\hat{p})^{2}\frac{p-(1-n)p^{2}}{n}$$ and plug in the estimate of $\hat{p}$ to get it.
