[R] generate random numbers that sum up to 1

[Alberto Monteiro]

I don't have the previous messages, but it seems to me that
the solutions didn't quite get the requirements of the problem.

For example, "it's obvious that" for n = 2, the
random variables X1 and X2 should be X1 <- runif(1)
and X2 <- 1 - X1; while the solution X <- runif(2);
X1 <- X[1] / sum(X); X2 <- X[2] / sum(X) will give
different distributions, as shown in this test case:

N <- 1000
M <- matrix(runif(2 * N), 2, N)
X <- M[1,] / (M[1,] + M[2,])
hist(X)

"It's obvious that" for a generic n-th dimensional
set of uniform variables X1, ... X_n subject to the
restriction X1 + ... + X_n = 1, the solution is to
take a uniform distribution in the (n-1)-th dimensional
hyperpyramid generated by the above relation and the
restrictions that each X_i >= 0.

For example, for n = 3, we should sample from the
equilateral triangle with vertices c(1,0,0), c(0,1,0)
and c(0,0,1).

For n = 4, we should sample from the pyramid whose
vertices are c(1,0,0,0), c(0,1,0,0), c(0,0,1,0) and
c(0,0,0,1).

I don't know if there is a simple formula to do this
sampling.

Alberto Monteiro