Compute expectation of certain $N$-th largest element of uniform sample A  premier B-school has 2009 students.The dean,a math enthusiast,asks each student to submit a randomly chosen number between 0 and 1.She then ranks these numbers in a list of decreasing order and decides to use the 456th largest number as a fraction of students that are going to get an overall pass grade this year.What is the expected fraction of students that get a passing grade?
I am not able to think in any direction. As it is really difficult to comprehend.

A naive approximation would be: Assume the $N=2009$ given numbers are equidistant within $[0,1]$. The the $k$th smallest should be about $\frac{k-\frac12}N$ and hence the 456th largest = 1554th smalles $\approx \frac{1544-\frac12}{2009}=\frac{63}{82}$. But this reasoning is a bit handwaving.
To be precise you should evaluate for $x\in[0,1]$:
What is the probability $p(x)$ that exactly 1553 guesses are $<x$ and exactly 456 guesses are $\ge x$?
Then the expected value we want is $\int_0^1 x p(x) dx$. Try this latter method for some smaller values of $N$ and $k$ and see how much it dieffers from the straightforward guess $\frac{k-\frac12}N$.
