Order statistics in the uniform (probability) distribution

[Mathmos6]

Homework Statement

Hi there: I just need someone to tell me if I've made a mistake somewhere in my solution to this:

Suppose that X1 , . . . , X2n+1 are i.i.d. random variables that form a random sample
from the U (0, 1) distribution. Suppose that the values are arranged in increasing order as
Y1 ≤ Y2 ≤ . . . ≤ Y(2n+1) . Calculate expressions for the distribution function and for the probability density function of the random variable Y(n+1) (the sample median).

The Attempt at a Solution

Now if we want Y(n+1) to be in the interval [a,b] we need to have exactly n of the Xi in [0,a] and n+1 in [a,1], but ensure not all of the latter n+1 are in [b,1]. So we have ${{2n+1}\choose{n+1}}a^n[(1-a)^n-(1-b)^n]$ where the $(1-a)^n-(1-b)^n$ ensures not all n+1 of the latter Yi are in the [b,1] interval. But shouldn't there be some sort of symmetry in a and b with this function: shouldn't it be F(b)-F(a) where F'(x) is the pdf of the question? Because it doesn't look much like an F(b)-F(a) to me...

Thanks a lot,

Mathmos6

 

[Mathmos6]

Anyone? Any help would be very much appreciated, no matter how small!