If I have a matrix $A \in \mathbb R^{N \times Y}$ which is uniformly distributed and with probability $\frac{1}{Y}$ gives the following value $\sqrt{\frac{Y}{N}}b_i$ where $b_i$ is the standard basis vector how do I go about calculating the variance of the following $L_2$ norm?
$$ \Vert A z_i - Az_j \Vert^2_2 $$
where $z_i$ and $z_j$ are column vectors. I'm just not sure how to extract the mean and variance of the matrix $A$ by using the properties of a uniform distribution. Any pointers?