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Re: Random points in triangle

  • To: mathgroup at smc.vnet.net
  • Subject: [mg111745] Re: Random points in triangle
  • From: Mark McClure <mcmcclur at unca.edu>
  • Date: Thu, 12 Aug 2010 05:28:28 -0400 (EDT)

On Sun, Aug 8, 2010 at 7:23 AM, Ray Koopman <koopman at sfu.ca> wrote:
> Try this a few times. It looks pretty uniform to me.
>
> p == RandomReal[NormalDistribution[0,1],{3,2}];
> r == #.p / Total[#,{2}] & @
>    RandomReal[ExponentialDistribution[1],{5000,3}];
> ListPlot[{p,r},PlotRange->All,AspectRatio->1,Frame->True,Axes->None,
>    PlotStyle->{{Red,PointSize[.02]},{Black,PointSize[.005]}}]

Yes, it turns out that it *must* be.  According to Wikipedia, the
obvious generalization works for the n-simplex:
http://en.wikipedia.org/wiki/Simplex#Random_sampling

If you are happy working with change of variables for triple integrals
and joint probability distributions, then the triangular case isn't
too hard.  Here's my proof:
http://facstaff.unca.edu/mcmcclur/blog/UniformTriangle.html

Mark McClure


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