Suppose that I have a random vector $X=(p_1, p_2, ..., p_n)$ where each $0\leq p_i\leq 1$ and $\sum p_i = 1$. I would like to find distribution for each component $p_i$ and also other general expressions such as $p_1+p_2+p_3+p_4$.
I have used function RandVec in R to generated one million vectors. Histograms for each $p_i$ can be seen here (for case $n=8$).
EDIT: Thank to @wuber, I have realized if I do not add any constraints, except sum = 1, then they follow Dirichlet(1, 1, ..., 1). Please correct me if this is incorrect.