Assume $X_1,X_2,\dots ,X_n \overset{i.i.d}{\sim} F(.) $
I define $X_{(1)},X_{(2)},\dots,X_{(n)} $ Like that: $X_{(1)}\leq X_{(2)}\leq \dots\leq X_{(n)}$
For $r\in \{1,2,\dots,n\}$ :
What is the formula of $E[X_{(r)}]$?
I know the distribution function of $X_{(r)}$
$F_r(x)=\sum_{i=r}^{n}\binom{n}{i} (F(x))^i(1-F(x))^{n-i} $
But I still don't know what should I do
i.i.d: independence identical distribution $F_r(x)$: distribution function of $X_{(r)}$
For continuous independence random variables $X_1,\dots,X_n \sim F(.)$ we can find $f_r(x)$ (density function of $X_{(r)}$):
$f_r(x)=\frac{1}{B(r,n-r+1)}[F(x)]^{r-1}[1-F(x)]^{n-r}f(x)$
Then we can write expectation like that: $E[X_{(r)}]=\int_{-\infty}^{+\infty} xf_r(x) dx$
For discontinuous random variable:
Assume $S^*_X=\{ x:f(x)>0\}$
We can write the $E[X_{(r)}]$ like that:
$E[X_{(r)}]=\sum_{x\in S^*_X}P(X_{(r)}\geq x) =\sum_{x\in S^*_X}1-F_r(x)$
B: Beta function