A distribution problem using Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg111459] A distribution problem using Mathematica
• From: "S. B. Gray" <stevebg at ROADRUNNER.COM>
• Date: Mon, 2 Aug 2010 07:04:37 -0400 (EDT)

```	I want 3 random variables v1,v2,v3 with uniform distribution from 0 to
1, but modified or normalized so that their sum is a uniform
distribution from 0 to 1.
These variables are for placing a point at a random place inside a
tetrahedron defined by 4 vertex vectors p1,p2,p3,p4. The internal point
is given by p=v1(p1-p4)+v2(p2-p4)+v3(p3-p4) (barycentric coordinates).
The 0 to 1 constraints assure that the point p will be inside.
The  closest I have come is this function giving the triplet
frs={v1,v2,v3}:

mul = RandomReal[{0, 1}, {3}];
mul = mul/Total[mul];
frs = Sqrt[RandomReal[{0, 1}, {3}]]*Power[mul, (3)^-1];

which I tested with the histogram nhist: (The +1 avoids trying to access
the 0th element of the list nhist.)

nhist = Table[0, {1000}];
Do [ mul = RandomReal[{0, 1}, {3}];
mul = mul/Total[mul];
frs = Sqrt[RandomReal[{0, 1}, {3}]]*Power[mul, (3)^-1];
ip  = IntegerPart[1000 frs][[2]];
nhist[[ip+1]]++, {10000}
];
Print[frs];
ListPlot[nhist]

This ad-hoc method gives a distribution that covers the range 0-1 but is
too heavy in the region 0.3 to 0.7. This would put too many points near
the middle of the tetrahedron. Something tells me there must be a better
and more elegant solution. Any ideas?

Steve Gray

```

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