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Topic: Random Triangle Problem
Replies: 57   Last Post: Aug 17, 1997 10:51 PM

 Messages: [ Previous | Next ]
 John Rickard Posts: 55 Registered: 12/6/04
Re: Random Triangle Problem
Posted: Aug 1, 1997 8:19 AM

Robert Hill (eclrh@sun.leeds.ac.uk) wrote:
: (4) x- and y- coordinates of each vertex independent normal N(0,1)
: (if my memory is correct this distribution is isotropic,
: i.e. for any fixed a it makes x cos a + y sin a also N(0,1)):
: 249791 249989 249974 250246

: Without running a statistical test, the results for (4) looked as though
: they might be compatible with my vague intuition that the theoretical
: answer for this case may be exactly 3/4.

It is. Angle BAC is obtuse iff A is closer to the midpoint of BC than
B and C are; that is, iff the vector A - (B+C)/2 is shorter than the
vector (B-C)/2. By the wonders of the normal distribution, if A, B, C
are independent with the same circularly symmetric 2-dimensional
normal distribution [is this the right terminology?], then A - (B+C)/2
and (B-C)/2 are independent with circularly symmetric 2-dimensional
normal distributions, the former having 3 times the variance of the
latter. A little calculus then shows that the probability that the
former has smaller magnitude is 1/4. (Is there any easy way to see
that this must be true without using calculus? If "3" is replaced
with "a", the probability becomes 1/(a+1).)

--
John Rickard

Date Subject Author
7/16/97 Mike Housky
7/21/97 Bill Taylor
7/22/97 tony richards
7/24/97 Brian M. Scott
7/23/97 tony richards
7/23/97 T. Sheridan
7/24/97 Bill Taylor
7/24/97 Bill Taylor
7/25/97 Ilias Kastanas
7/23/97 Robert Hill
7/23/97 tony richards
7/27/97 Bill Taylor
7/24/97 Robert Hill
7/28/97 tony richards
7/30/97 Bill Taylor
7/30/97 tony richards
8/1/97 Bill Taylor
7/24/97 Robert Hill
7/24/97 Robert Hill
7/24/97 Robert Hill
7/25/97 Robert Hill
7/30/97 Bill Taylor
8/1/97 Charles H. Giffen
8/1/97 John Rickard
8/1/97 Chris Thompson
8/1/97 John Rickard
8/4/97 Bill Taylor
8/5/97 John Rickard
7/25/97 Charles H. Giffen
7/25/97 Charles H. Giffen
7/28/97 Hauke Reddmann
7/28/97 Robert Hill
7/28/97 Robert Hill
7/28/97 Robert Hill
7/29/97 tony richards
7/30/97 Keith Ramsay
7/30/97 tony richards
8/2/97 Keith Ramsay
7/29/97 tony richards
8/4/97 Bill Taylor
8/5/97 Charles H. Giffen
8/6/97 Terry Moore
8/7/97 Terry Moore
8/16/97 Kevin Brown
8/17/97 Kevin Brown
7/30/97 Robert Hill
7/31/97 tony richards
8/6/97 Terry Moore
7/31/97 John Rickard
7/30/97 Robert Hill
7/31/97 Robert Hill
7/31/97 Robert Hill
8/1/97 R J Morris
8/4/97 Robert Hill
8/4/97 Robert Hill
8/5/97 Charles H. Giffen
8/6/97 Robert Hill