random vector v from uniform distribution at (0,1) with sum(v)=1

Hello,

How can I generate a uniformly distributed random vector with its sum to be equal to 1?

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Too many people think that generating a uniform sample, then normalizing by the sum will generate a uniform sample. In fact, this is NOT at all true.

A good way to visualize this is to generate that sample for the 2-d case. For example, suppose we do it the wrong way first?

xy = rand(100,2);
plot(xy(:,1),xy(:,2),'.')

Now, lets do the sum projection that virtually everyone poses. (Yes, it is the obvious choice. Now we will see why it is the wrong approach.)

xys = bsxfun(@rdivide,xy,sum(xy,2));
hold on
plot(xys(:,1),xys(:,2),'ro')
axis equal
axis square

The sum-projected points lie along the diagonal line. Note the distribution seems to be biased towards the middle of the line. A uniform sample would have points uniformly distributed along that line.

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