I have an application where I need to pick a probability distribution [itex](x_{1},\ldots,x_{n})[/itex] at random and uniformly from the simplex of all points for which the coordinates add up to 1, i.e. [tex]\sum_{i=1}^{n}x_{i}=1.[/tex] Surprisingly, I didn't find much about simplex point picking on the internet, but http://en.wikipedia.org/wiki/User:Skinnerd/Simplex_Point_Picking appears to address this issue. Skinnerd suggests to pick individual members of [itex](y_{1},\ldots,y_{n})[/itex] randomly from a uniform distribution over the interval [itex](0,1)[/itex] and then take [tex]x_{i}=\frac{\ln{}y_{i}}{\sum{}\ln{}y_{i}}.[/tex] So far so good (although, why does he need the minus sign in his [itex]x_{i}=-\ln{}y_{i}[/itex]?).(adsbygoogle = window.adsbygoogle || []).push({});

My question is: what is the distribution of [itex]x_{i}[/itex] over the interval [itex](0,1)[/itex], i.e. what is the probability [itex]P(a<x<b)[/itex] that one of these coordinates is in [itex](a,b)\subseteq{}(0,1)[/itex]?

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# Simplex point picking

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