Prove Corollary 4.4. 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 \(F\) has density \(f\) in a neighborhood of \(\xi_{p}\) and that \(f\) is positive and continuous at \(\xi_{p}\). Then, prove that

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

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

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Corollary 4.4. Let {X} be a sequence of independent random variables that have a common distribution F. Let p = (0, 1) and suppose that F has density f in a neighborhood of Ep and that f is positive and continuous at Ep. 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.