Suppose we have $D$ random variables $X_1, \ldots, X_D$, and $X_i\in[0,1]$ for $i=1,\cdots,D$. Can we somehow design a joint distribution $P(X_1, \ldots, X_D)$, such that $\sum_{i=1}^{D}X_i=s$, where $s\in[0, D]$, always holds, with some assumptions? Preferably there are some hyperparameters $\alpha$ in $P(X_1, \ldots, X_D;\alpha)$ that can control the concentration of $X_i$. A possible solution is the Dirichlet, however it assumes that $\sum_{i=1}^{D}X_i=1$ instead of any $s\in[0,D]$.
One practical example could be: we want to send $s$ dollars cashback to $D$ customers, but each customer can receive at most 1 dollar. How can we design a distribution to allocate the money randomly?
I thought this should be a common problem in many applications, but I failed to find any useful information.