Suppose I draw $N$ random variables from independent but identical uniform distributions, where $N$ is an even integer. I now sort the drawn values and find the two middlemost of these. Finally, I calculate a simple average of these two middlemost values.
Has anyone, to the reader's best knowledge, examined the progression of distributions that arise as N increases from $N = 2$ to $N = \infty$ ? The first distribution is easily found to be Triangular, but what about the rest? Plots from simulations in MATLAB, with a uniform distribution on the range $0$ to $1$, provide the following illustrations:
I think that this converges to 0.5 in probability, assuming that the distributions are all $Unif(0, 1)$. I cannot provide a rigorous derivation, but my logic is that as $n \rightarrow \infty$ the interval between the two points shortens and moves and is ever more likely to contain 0.5.