If $X_1,...,X_n$ are iid random variables with $r$th order statistic $X_{(r)}$ I'm trying to prove it's pdf is
$f_{(r)}(x)=\frac{n!}{(r-1)!(n-r)!}F(x)^{r-1}[1-F(x)]^{n-r}f(x)$
Is is true for $r=1$. I then assume true for $r$. The only step in this proof is I don't understand why this assumption implies that the cdf for $X_{(r)}$ is:
$F_{(r)}(x)=\sum_{j=r}^n {n\choose j} F(x)^{j}[1-F(x)]^{n-j}$
Is someone could explain this I would be very grateful