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If I have two independent uniforms $U_1$ and $U_2$, one with parameter $(a_1, a_2)$, the other one with parameters $(b_1, b_2)$ and I want to find out the variance of $U = U_1 + U_2$, I use

$Var(U) = Var(U_1) + Var(U_2) +Cov(U_1,U_2) = \frac{(a_2-a_1)^2}{12} +\frac{(b_2-b_1)^2}{12} +0$

Is it sound to assume the covariance is 0, since the the R.V.'s are independent, and therefore uncorrelated, or may I not assume this?

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    @Alex: Makes se$n$se. thanks2012-10-26

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As Alex put it in the comments:

If RVs $X, Y$ are independent, then $\operatorname{corr}(X,Y) = \sqrt{\dfrac{\operatorname{cov}(X,Y)}{\operatorname{Var} X \operatorname{Var} Y}} = 0$. The opposite is not necessarily true, i.e. if $\operatorname{cov}(X,Y) = 0$ RVs may not be independent. Your solution is correct since RVs are independent.

See Wikipedia for an example where uncorrelated variables are not independent.