Discussion
Let \(F_{n}(\cdot)\) be empirical cumulative distribution function, based on a random of size \(n\) from a continuous distribution with cumulative distribution function \(F(\cdot)\). Then :
Answer: C
The asymptotic distribution of \(F_{n}\) is normal.
At a fixed \(x\), \(nF_{n}(x)\) counts the observations not exceeding \(x\), which is binomial with parameters \(n\) and \(F(x)\). By the central limit theorem, \(\sqrt{n}(F_{n}(x) - F(x))\) tends to \(N(0, F(x)(1 - F(x)))\).
The other statements fail. \(F_{n}(x) \to F(x)\) almost surely (and uniformly, by Glivenko-Cantelli), so \(F_{n}\) is consistent. Its variance is \(F(x)(1 - F(x))/n\), which tends to 0, not to a positive constant. And its mean is \(F(x)\), not 1/2.
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