Conditional distribution of the sum of k, independent ordered draws

Question

Suppose I make k independent draws from the same distribution (say uniform). The distribution of the sum of these order statistics should equal the distribution of the sum of k independent random variables, am I right? But what if I am also told that the highest of these draws has a value less (or equal than) x? In other words, what is the expression for the distribution of the sum of k order statistics conditional on the highest of them not exceeding x?

Answer 1

If you know that the highest draw is at most $x$, then you know that all $k$ draws are at most $x$, and otherwise they are arbitrary. So in effect you are sampling from your distribution conditioned on being at most $x$.

For example, if your initial distribution is uniform on $[0,1]$ and $x \in [0,1]$, then the expected sum of $k$ independent draws conditioned on the maximum being at most $x$ is the same as the expected sum of $k$ independent draws of a uniform $[0,x]$ variable, which is $kx/2$.