Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\). Let \(\theta\) be defined as the \(p^{\text {th }}\) quantile of \(F\).

a. Write \(\theta\) as a functional parameter \(T(F)\) of \(F\).

b. Develop a plug-in estimator for \(\theta\) based on using the empirical distribution function to estimate \(F\).

c. Consider estimating \(F\) with a \(\mathrm{N}\left(\bar{X}_{n}, Sight)\) distribution. Write this estimator in terms of \(z_{p}\), the \(p^{\text {th }}\) quantile of a \(\mathrm{N}(0,1)\) distribution.