Moment method estimator of binomial population

Question

If question asked... Let X~Bin(n,$\pi$). Find the methods of moments estimator MME of $\pi$. Book answer was $\pi$_hat=X_bar/n

Shouldn't it be X_bar/n_hat , or even better 1 -$\sum(X_i-Xbar)^2$/$\sum X_i$? by solving the 2 equations E(x)=$M_1 and Var(x)=M_2-(M_1)^{2} , M_i=\sum X_i^{2}$/n Thanks

Answer 1

There is only one parameter to estimate, $\pi$.

Note that $n$ is known.

Hence from the first moment, $$\bar{X} = n\hat{\pi}$$

Hence $$\hat{\pi}= \frac{\bar{X}}{n}$$