# pr.probability – Projection of a random vector distributed uniformly on the cube

Let $$Ainmathbb{R}^{mtimes k}$$ be a fixed matrix with $$m, and let $$X$$ be a random vector distributed uniformly on the $$k$$-cube $${-1,1}^{k}$$.

What is the distribution of $$Y=AX$$?

Denoting by $$a_{i}^{top}$$ the $$i$$th row of $$A$$, we can see that the $$i$$th entry of $$Y$$ can take (at most) $$2^{k}$$ possible values from $${a_{i1}+cdots+a_{ik},ldots,-a_{i1}-cdots-a_{ik}}$$, and each of these is equally likely. How do I compute the full joint distribution? Any hints appreciated.

(Also posted on Math StackExchange)