Let $X$ be a non-negative random variable distributed on $[a,b]$ where $0<a<b<+\infty$ with CDF $F(x)$ and density function $f(x)$. Let $X_{1:n}$ be the first order statistic of $ n\geq 2$ independent variates each with CDF $F(x)$.
Is there a general relationship between the expected value of $X_{1:n}$ and that of $X$?
Surely, $E(X)= \int_a^b x f(x) dx$ and it's easy to see that $E(X_{1:n})=n \int_a^b [1-F(x)]^{n-1} x f(x) dx$.
Alternatively, $E(X)= \int_a^b (1-F(x))dx$ and that $E(X_{1:n})= \int_a^b (1-F(x))^n dx$.
Is there a way so express $E(X_{1:n})$ in terms of $E(X)$ in general?