Let we have $X_1 ... X_n$ - iid random variables with $F(x)$ - distribution function.
So we have $X1:n$ and $Xn:n$ - minimum and maximum.
Help me please with counting conditional expectation $E(Xn:n|X1:n=x$) in terms of $F(x)$ ?

If we have the joint distribution we can find the conditional distribution of the max conditioned on the min. If we have the conditional distribution we can find the expected value.

So it looks like if you were to keep everything just in terms of F it would look fairly messy - but the idea doesn't seem too bad.