Emiprical cdf expected value and variance

Question

Let $X_1, \ldots, X_n$ be independent random variables of identical uniform distribution $[0,1]$. Let $F(t)$ denotes empirical cdf at point t (estimated on the basis of observed values $X_1,X_2, \ldots, X_n$.

Find $\mathbb{E}F(1/4)$ and $Var F(1/4)$

My attempt: $EX=1/4$ (variance similarly, but i am not certain about that). My guess for variance would be $Var 1/4=0$

Am i correct?

Answer 1

$\Pr\left(F(t) = \frac{k}{n}\right) = {n \choose k}t^k(1-t)^{n-k}$ which is essentially a scaled binomial distribution, for $0 \le k \le n$.

$nF(t)$ is binomial with parameters $n$ and $t$, and so has mean $nt$ and variance $nt(1-t)$

so $F(t)$ has mean $t$ and variance $\frac{t(1-t)}{n}$

Answer 2

Hint:

If I understand the concept of empirical CDF correctly then:$$F(t)=\frac1{n}\sum_{i=1}^n 1_{(-\infty,t]}(X_i)$$

By linearity of expectation and symmetry find $\mathbb EF(t)$ and (with extra aid of independence) $\text{Var} F(t)$.

Then substitute $t=\frac14$.