Prove Corollary 4.3. That is, let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables that have a common distribution \(F\). Let \(p \in(0,1)\) and suppose that \(F\) is differentiable at \(\xi_{p}\) and that \(F\left(\xi_{p}ight)>0\). Then, prove that

\[\frac{n^{1 / 2} F^{\prime}\left(\xi_{p}ight)\left(\hat{\xi}_{p n}-\xi_{p}ight)}{[p(1-p)]^{1 / 2}} \xrightarrow{d} Z\]

as \(n ightarrow \infty\) where \(Z\) is a \(\mathrm{N}(0,1)\) random variable.

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Corollary 4.3. Let {X} be a sequence of independent random variables that have a common distribution F. Let p E (0, 1) and suppose that F is differentiable at Ep and that F(Ep) > 0. Then, n1/2 F' (Ep)(pn-p) d Z [p(1 - p)]1/2 as no where Z is a N(0, 1) random variable.