Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 (Tek) 9/26/83; site tekchips.UUCP Path: utzoo!watmath!clyde!floyd!harpo!decvax!ucbvax!ucbcad!tektronix!tekchips!stevev From: stevev@tekchips.UUCP Newsgroups: net.math Subject: Re: Yet another probability puzzle Message-ID: <589@tekchips.UUCP> Date: Mon, 27-Feb-84 15:41:28 EST Article-I.D.: tekchips.589 Posted: Mon Feb 27 15:41:28 1984 Date-Received: Wed, 29-Feb-84 12:38:07 EST Organization: Tektronix, Beaverton OR Lines: 50 > 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 leftmost and rightmost 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 h(y) 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. 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 The expected value of h(y) (which was the original question) is then found easily by integrating y h(y) dy over the interval [0,1], giving