Perhaps is not so hard but it's been a while since I did any statistics. Say I have two random variable matrices $M,V$ of dimensions $l\times m$ and $m\times 1$ resp. such that each entry has uniform distribution $U(0,a)$. What is the distribution of $MV$?
My guess is that it is enough to sort out what happens when $M$ is just a vector with uniform entries, then, depending on the value of $m$. If, say $m\ge30$, using central limit theorem we have that the distribution of $MV$ is normal with appropriate parameters, namely:
Let $M=(X_i)_1^m$ and $V=(Y_i)_1^m$ with all the variables iid and $U(0,a)$ for some $a\in\mathbb{R}^+$, then $$MV=\sum X_iY_i\sim\mathcal{N}(ma/2,ma^2/12)$$
is this the right approach?