If f is a pdf of a real-valued random variable X, the cumulative distribution function (cdf) of X is F : R â [0, 1] so that F(x) = $$\int_{-\infty}^x f(t) dt$$ for all x. For an event A â R, let 1[A] be 1 if A is true and 0 otherwise. Let X1,..., Xn be n samples of X that are independent and identically distributed with cdf F. For all x, define $$\hat{F}_n(x) = \frac{1}{n} \sum_{i=1}^n 1[X_i \le x]$$ known as an empirical estimate of F(x).
(a) For any x â R, 1[X ⤠x] is 1 when X ⤠x and 0 otherwise. Show that F(x) = E[1[X ⤠x]].
(b) For any x and n, what is E[$\hat{F}_n(x)$]?
(c) For any x, show that the variance of Fâ(x) is Var(Fâ(x)) = F(x)(1 â F(x)).
(d) For any x and n, show that the variance of $\hat{F}_n(x)$ is $\frac{1}{n}$F(x)(1 â F(x)).
(e) For any x and n, show that E[($\hat{F}_n(x)$ - F(x))²] ⤠$\frac{1}{4n}$.
