In Matlab the code
v = rand(1,3); v = v/sum(v);
is sometimes suggested as a convenient means of generating three random variables, whose ranges are restricted to [0,1], which have a fixed sum of one. However, this procedure has the property that the area-wise density distribution of the three values, considered as cartesian coordinates in 3D space, is widely variable throughout the planar region of possible locations of v. For any given density value in the range of this density distribution, let A be the corresponding area of the subregion of all points whose density is less than or equal to this given value, and let P be the corresponding probability that v would lie in this subregion. The task is to write a function 'fixedsumarea' which receives P as an input and gives A as an output:
A = fixedsumarea(P);
You should assume that initially 'rand(1,3)' perfectly generates three independent random variables each uniformly distributed on [0,1], but subsequently each is modified by being divided by their mutual sum.