For the case of more generic sum of n variables Xi is equal to 1.
If the apriori distribution is supposed to be uniform density
Xi ~ U(0,1)
If the conditioning is
sum(Xi) = 1
we can show that the Xi conditional probability distribution has density of
pdf(Xi | sum Xi = 1) = (n-1)*(1-x)^(n-2).
Note that for n=2, the conditioning distribution happens to be a uniform distribution. This is not true of n > 2.
One neatly way to generate a proper conditioning distribution is using n independant exponential distribution then normalize by the sum
n = 4;
Y = -log(rand(100000,n));
X = Y ./ sum(Y,2); % Each row is a realization of the conditining
We can check it gives the expected conditioning distribution
x=linspace(0,1);
pdf=@(x)(n-1)*(1-x).^(n-2);
histogram(X(:),'Normalization','pdf');
hold on
plot(x,pdf(x),'MarkerSize',3);
title('exponential normamlization');
If you want further to constraint the Xi with in a lower and upper bounds, Roger Stafford's randfixedsum function does the job.
