In article <33E254FD.7B33@virginia.edu>, "Charles H. Giffen" <email@example.com> writes: > Bill Taylor wrote: > > > > firstname.lastname@example.org (Robert Hill) writes of some fascinating simulation data! > > > > |> I generated 1 million pseudo-random triangles > > > > |> (1) vertices independent uniform inside a square: > > |> obtuse at first vertex at second at third all acute > > |> 241781 241998 241802 274419 > > |> > > |> (2) vertices independent uniform inside a circle: > > |> 248534 248198 248619 254649 > > |> > > |> (3) vertices independent uniform inside an equilateral triangle: > > |> 249377 249031 249279 252313 > > > > These are all *hugely* away from 3/4 to 1/4. The best has a chi-square > > of about 20 for 1 df. MADLY significant. > > > > [snip] > > Hm! Out of 10 million triangles, I got > > 7483180 obtuse inside an equilateral triangle > 7250536 obtuse inside a square > 7196418 obtuse inside a circle > 9079940 obtuse inside a rectangle (1 x 4, I believe). > > (vertices independent inside the region in question). > > How good is your random number generator (mine generates random > 32 bit integers and then rescales -- and the algorithm is one > recommended by Knoth).
Don't blame the subtleties of my random number generator (NAG Fortran Library G05CAF with 64-bit IEEE reals on Sun and SGI machines), blame my gross programming mistake!
Your figures for a square and an equilateral triangle seem reasonably close to mine. If I remember rightly we should expect differences of a few times sigma = sqrt(npq) = about 430 on samples of 1 million.
My program for a circle contained a stupid error. I was trying to generate uniform points in a square and reject those outside the inscribed unit circle, but I forgot to first transform from the square [0,1]x[0,1] to [-1,1]x[-1,1], so those figures are actually for a quadrant. Correcting the bug, I get 720578 obtuse out of a million, or 7200949 out of 10 million, reasonably close both to your figures and to the theoretical .7187 that has now been posted by Keith Ramsay.
This makes the results for a square intermediate between those for a triangle and those for a circle, destroying of a surprising observation I made.
-- Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, the heart is true." - John Dowland, Fine Knacks for Ladies (1600)