How to generate a random data vector that follows a constraint

I want to generate random data vector where that follows this constrainst. where and is a constant. I would appreciate any help.

Answer 1

[Bruno Luong]

This will generate uniform distribution (within to polytope)

N = 10;
Etotal = 0;

E = -log(rand(N+1,1));
E = E(1:N,:)./sum(E,1);

[Bruno Luong]

Here is a comparison of distribution with the two other methods proposed below to show the issue of non-uniformity if one doesn't pay attention

N = 2;
Etotal = 1;
p = 3e3;

Ebias = rand(1,p) .* randfixedsum(N,p,Etotal,0,Etotal);
E = rand(1,p).^(1/N)*Etotal .* randfixedsum(N,p,1,0,1);
E2 = -log(rand(N+1,p)); E2 = E2(1:N,:)./sum(E2,1);

subplot(2,2,1)
plot(Ebias(1,:),Ebias(2,:),'.');
title('rand *  randfixedsum')
axis equal

subplot(2,2,2)
plot(E(1,:),E(2,:),'.');
axis equal
title('sqrt(rand) *  randfixedsum')

subplot(2,2,3)
plot(E2(1,:),E2(2,:),'.');
axis equal
title('exponential method')

[Saifullah Khalid]

Bruno Luong, thank very much for detailed answer and the insight.

Answer 2

[Walter Roberson]

Look in the File Exchange for Roger's randfixedsum(). Generate a vector with fixed sum . Multiply all of the elements by rand() to implement the <= part. (Though you might want to worry about the difficulty that rand() is never exactly 1, so if you generate a sum exactly equal to and multiply by rand() then the result can never exactly total

[Bruno Luong]

IIUC you propose

E = rand()*Etotal * randfixedsum(N,1,1,0,1) 

this will generate bad distribution (too many point close to 0 compare to uniform distribution), espetially for N >> 1.

[Walter Roberson]

I was thinking of

rand() * randfixedsum(N,1,Etotal,0,Etotal)

but your comment might still apply.

[Bruno Luong]

Both give the same non-uniform pdf

The "correct" one is

E = rand()^(1/N)*Etotal * randfixedsum(N,1,1,0,1);