I'm trying to formulate a way to generate a random set of 4 numbers that sum to 1 (a discrete probability distribution) such that each is uniform, ~$U(0,1)$. They obviously can't be iid, and normalized uniforms have a mean 0.25 which is way too low for our purposes. The way I've been trying to tackle it is by giving each one a 25% chance of coming from a "bigger" distribution.
Let $X_1,X_2,X_3$ iid ~$U(0,1)$ and $Y$ with support $(0,\infty)$. Let $A={X_1\over X_1+X_2+X_3+Y}, B={Y\over X_1+X_2+X_3+Y}$. Find the $F_Y$ such that ${1\over 4}f_B(z)+{3\over 4}f_A(z)=1.$
The mean of each of them will have to be $0.25$ if they come from the same distribution because means add. The easiest approach is to pull four numbers from $U(0,1)$ and normalize them by dividing by the sum. This will give each a distribution which ranges all the way up to $1$, but there will be little density there.