How do we sample uniformly at random from the unit dimensional simplex? It turns out there is an easy way to do it: generate i.i.d. uniform variables in [0,1] and let denote the k-th smallest among them. Then,   is a desired sample from the unit simplex . It is straight-forward to prove now the claim and implement the method. However, there is a more efficient approach, based on the following exercise:

Exercise 1: Let be i.i.d. exponential random variables and define . Prove that is uniformly distributed in the dimensional unit simplex .

As this paper explains in detail, the former method time whereas the second . Here is also a visualization of samples from the 2-simplex using a simple script.