I am wondering if there exists a distribution of $k$ positive random variables and their sum is a given deterministic scalar, say $m$. Mathematically, I am finding the following distribution: $$ \mathbb{P}(X_1,\cdots,X_k),\quad s.t. \quad X_1+\cdots+X_k=m. $$ I am hoping that this distribution has a parameter which is like the variation parameter $\sigma^2$ in Gaussian distribution, that can characterize the variation of the $k$ random variables.
My thoughts
Entropy might be a good metric to describe the variation of such $k$ random variables. However, I do not know how to generate those $k$ random variables that satisfy their summation being $m$, and their entropy being, say $h$.
This is just my thoughts. If there are any similar distributions and have the desired parameter, please let me know. Thanks ahead!
Something like a symmetric Dirichlet distribution, i.e. with equal concentration parameter, may be the kind of thing you are looking for.
You can generate them as $k$ Gamma random variables and then divide by their sum. That would then add up to $1$, so you may want to multiply by $m$.