Value of $c$ that makes $c(\bar X (1 - \bar X))$ an unbiased estimator of $p(1 - p)$ I need to find the value of $c$ that makes $c(\bar X (1 - \bar X))$ an unbiased estimator of $p(1 - p)$.
I am confused about how to do this. I want to say that the value of $c$ must be $1$ because $E(\bar X) = \frac{\mu}{n} = \frac{np}{n} = p$, but that seems too easy.
Any advice? Thanks in advance.

$$
\operatorname{var}(\overline X) = \frac{p(1-p)} n
$$
So
$$
\operatorname E(\overline X^2) = p^2 + \frac{p(1-p)} n
$$
$$
\operatorname E(\overline X(1-\overline X)) = p - p^2 - \frac p n + \frac {p^2} n = \left( 1 - \frac 1 n \right) p(1-p).
$$
So the value of $c$ that you need is the reciprocal of $1 - \dfrac 1 n.$
