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From: stevev@tekchips.UUCP (Steve Vegdahl)
Newsgroups: net.math
Subject: Re: Yet another probability puzzle
Message-ID: <590@tekchips.UUCP>
Date: Mon, 27-Feb-84 16:04:25 EST
Article-I.D.: tekchips.590
Posted: Mon Feb 27 16:04:25 1984
Date-Received: Wed, 29-Feb-84 07:43:04 EST
Organization: Tektronix, Beaverton OR
Lines: 64

> What is the expected value of the range of N random points on
> a line from 0 to 1?  I think that's a concise statement of the
> problem.  To avoid ambiguity (I'm not a mathematician), I'll
> restate it as I conceived it: You have this (finite) 1-dimensional
> dart board at which you throw random darts.  What is the
> expected dispersion of N darts if they must all hit the board,
> but any point within is equally probable?

Assuming that your definition of "range" and "dispersion" is the
distance between the min and max points on the line, I believe that
answer is (N-1)/(N+1).

Here is my reasoning (this was all conjured up without any stat or caluculus
books, so someone please post a note if I goofed up).  The density function
for the max of N uniformly distributed random varibles on [0,1] is

			  N-1
		f (x) = Nx
		 N

The density function for the min of N uniformly distributed random varibles
on [0,z] is
			   N	   N-1
		g (x,z) = --- (z-x)
		 N	    N	
			   z

The density function for the distance between min and max computed by
integrating over combinations of points whose difference is y.

			 / 1
		        |
		h (y) = |  f (x) g  (x-y,x) dx
		 N      |   N     N-1
		       / y

(Please excuse the "ascii" integral sign.)  This integral basically sums the
probablities of the max of the N points being at x, and the min of the
remaining N-1 points (which are now limited to being <= x) being at x-y.
(Note that random variables for the min and max points are not independent,
so it doesn't work to compute their expectations independently and then
subtract.)  This integration is easy to do because lots of terms cancel out.
the result is
				    N-2
		h (y) = N(N-1)(1-y)y
		 N

The expected value can then be computed by integrating

		y h (y) dy
		   N

over the interval [0,1], giving our result of (N-1)/(N+1).

As I said before, I don't have a lot of time to verify this.  A sanity check
however, indicates that it works for N = 1, where the range should obviously
be zero, and approaches 1 as N approaches infinity, again consistent with
intuition.  Finally, the density function h integrated over [0,1] is 1, and
is clearly always non-negative for positive N, hence it is a feasible density
function.  Would someone like to corroborate or contradict?

				Steve Vegdahl
				Tektronix Inc.
				Beaverton, Oregon