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.


closed as off-topic by Matt F., RP_, user44191, Ilya Bogdanov, Jochen Glueck Sep 11 at 20:41

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    $\begingroup$ This is too trivial for this site. More appropriate for math.stackexchange.com or stats.stackexchange.com . Anyhow, Var($X_i$) = $c^2/12$. So $Var(Y) = \Sigma_{1=1}^n\alpha_i^2 Var(X_i)= c^2/12$. $\endgroup$ – Mark L. Stone Sep 11 at 16:36
  • $\begingroup$ Mark, thank you. I see now that you are correct (both in your answer and the assessment of the question). Somewhat surprising to a non-statistician like me, given the ugly form of the density for such random variables. $\endgroup$ – Dave Johannsen Sep 11 at 17:43