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For $K\ge 2$ and parameters ${\alpha }_i>0$, the Dirichlet distribution is a probability distribution on the probability simplex $p_i\ge 0$, ${\sum }_{i=1}^K{p}_i=1$. With ${\alpha }_0={\sum }_i{\alpha }_i$, its probability density function in coordinates $p_1,\dots ,p_{K-1}$, with $p_K=1-{\sum }_{i<K}{p}_i$, ison the simplex interior. This is a density relative to $d{p}_1\cdots d{p}_{K-1}$, not to $K$-dimensional Lebesgue measure, for which the simplex has zero volume.^[ref]

For $K=2$, the first coordinate has the beta distribution with parameters ${\alpha }_1,{\alpha }_2$. For $K=3$ and all parameters one, the coordinate triangle has area $1/2$ and density $2$, giving the uniform law in those coordinates. More generally, independent gamma distributions with shapes ${\alpha }_i$ and a common rate can be normalized by their sum to generate a Dirichlet vector. The expected value of $p_i$ is ${\alpha }_i/{\alpha }_0$; the parameters describe both average proportions and their concentration, rather than allowing the coordinates to vary independently.