Discussion
Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid observations from a continuous distribution F. Define :
\[ F_{n}(x) = \begin{cases} 0 & x \lt X_{(1)} \\ k/n & X_{(k)} \le x \lt X_{(k+1)} \\ 1 & x \ge X_{(n)} \end{cases} \]
Then for any fixed \(x\), \(\sqrt{n}(F_{n}(x) - F(x)) \xrightarrow{d} Z\), where Z is a normal r.v. with mean 0 and variance :
Answer: D
The variance is \(F(x)(1 - F(x))\).
For fixed \(x\), \(nF_{n}(x)\) counts the observations at most \(x\), so it is binomial with parameters \(n\) and \(F(x)\). Hence \(E[F_{n}(x)] = F(x)\) and \(\mathrm{Var}[F_{n}(x)] = F(x)(1 - F(x))/n\).
By the central limit theorem, \(\sqrt{n}(F_{n}(x) - F(x)) \to N(0, F(x)(1 - F(x)))\).
The other variances, \(F^{2}/2\), \((1 - F)^{2}\) and \(F^{2}\), are not the Bernoulli variance \(p(1 - p)\) with \(p = F(x)\).
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