$($a$)$

$3$ points possible $($graded$)$

$x_{i}Ber(p),$pin$(0,1)$

$($Enter barX$\_$n for the sample average ${\stackrel{}{x}}_n$

${\stackrel{}{x}}_n=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i.$

Maximum likelihood estimator hat$(p)=$

${\stackrel{}{x}}_n$

Hint: Use the definition of Fisher information that leads to the shorter computation.

$($If the Fisher information is not defined, enter DNE.$)$

Fisher information $I(p)=$

Use Fisher Information to find the asymptotic variance $V(hat(p))$ of the MLE hat$(p).$

$V(hat(p))=$