View Question
Q: probability theory ( No Answer,   1 Comment )
 Question
 Subject: probability theory Category: Science > Math Asked by: leosong-ga List Price: \$3.00 Posted: 15 Nov 2006 16:46 PST Expires: 15 Dec 2006 16:46 PST Question ID: 783088
 ```Let X(1), X(2),... X(2n+1) be independent identically distributed U[0,1] random variables. Define M(n) is the median them.show that M(n) converges to 1/2 in probability and almost surely .```
 ```Let's derive the pdf for M(n). (I'll speak loosely here - when I say "at x", I mean "between x and x + dx", where dx is an infinitesimal and so forth and we handle things properly. I'll also speak of the RVs having a given value; I mean a particular observation of them. All this is to save space and typing, because I'm lazy.) For M(n) to be at x, we need n individual X_i to be below x, n above it and the middle one "equal" to it. So, using the notation (a, b) = a choose b, the pdf is going to be f(x) = (2n+1, n).x^n . (n+1).(1-x)^n . 1 since there are (2n+1, n) ways of selecting the X_i to be below x, each has a probability x of being below x, (n+1, n) = n+1 ways of choosing which of the remaining X_i are to be above x, and each of those has probability (1-x) of being above x; 1 is the pdf of the final X_i at the point x. Clearly this is symmetrical about x = 1/2. It's easy to show that it has a maximum at x = 1/2 and minima at x = 0, 1. From here you should be able to use approximations for large n to put some bounds on the probability that M(n) will be more than a given distance from 1/2, and show that this goes to 0 as n goes to infinity.```