I have three random real numbers: x1, x2 and x3
Each has the constraint x > 0.1
And together they follow the constraint x1 + x2 + x3 = 1
I want to simulate a uniform distribution of all possibilities of (x1,x2,x3)
My current thought is as follows:
With the latter constraint, the df becomes 2 - let's consider x1 and x2
We can use a system of inequalities to represent the above constraints:
0.1 < x1 < 0.8
0.1 < x2 < 0.8
x1 + x2 < 0.9
which forms a right-angled isosceles triangle on the Cartesian plane.
I plan to simulate [x1~U(0.1,0.8),x2~U(0.1,0.8)]. For all resultant points not satisfying inequality #3, they are reflected along x1 + x2 < 0.9 (or, alternatively, discarded).
This should result in a uniform distribution for the triangular area.
My question is whether this is the most efficient algorithm. Can I somehow modify the simulation to simulate the triangle directly? i.e. eliminating the need to do the reflection or to discard any simulated point.