Suppose that $\{X_i\}$, $1 \le i \le n$ are i.i.d. uniform random variables, defined on the interval $[-c/2, c/2]$, $c \in \Re^+$. Let $\{\alpha_i \mid \alpha_i \in \mathbb{R}\}$ be s.t., $\sum_{i=1}^n \alpha_i^2 = 1$ (i.e., $\alpha = (\alpha_1, \ldots, \alpha_n) \in S^{n-1}$). Finally, let $Y=\sum_{i=1}^n \alpha_i X_i$, be the weighted sum of the $\{X_i\}$. Is there a convenient (i.e., closed-form) representation of the variance of $Y$? Should I know much about $Y$, other than guessing that we must have $\mathbb{E}(Y) = 0$?

Sadly, I've had almost no training in statistics, but would like to understand something about the properties of a certain class of random vectors for my work.