Let \(Y\) be a Bernoulli random variable with success probability \(\operatorname{Pr}(Y=1)=p\), and let \(Y_{1}, \ldots, Y_{n}\) be i.i.d. draws from this distribution. Let \(\hat{p}\) be the fraction of successes (1s) in this sample.

a. Show that \(\hat{p}=\bar{Y}\).

b. Show that \(\hat{p}\) is an unbiased estimator of \(p\).

c. Show that \(\operatorname{var}(\hat{p})=p(1-p) / n\).