# Variance of weighted sums of uniform variables [closed]

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 GlueckSep 11 at 20:41

This question appears to be off-topic. The users who voted to close gave these specific reasons:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Matt F., Ilya Bogdanov, Jochen Glueck
• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – RP_, user44191
If this question can be reworded to fit the rules in the help center, please edit the question.

• 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 $Var(Y) = \Sigma_{1=1}^n\alpha_i^2 Var(X_i)= c^2/12$. – Mark L. Stone Sep 11 at 16:36
• 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. – Dave Johannsen Sep 11 at 17:43