Question: 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

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

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