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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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    If RVs $X, Y$ are independent, then $corr(X,Y) = \sqrt{\frac{cov(X,Y)}{Var X, Var Y}}=0$. The opposite is not necessarily true, i.e. if $cov(X,Y)=0$ RVs may not be independent. Your solution is correct since RVs are independent.2012-10-26
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    Whether you *may* assume that they're independent depends on the context, which we don't know. You certainly *did* assume it at the outset, so you can use that assumption and conclude that the variables are uncorrelated.2012-10-26
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    @Alex: thanks (worthy of an answer IMO). Why can't we assume they are independent if cov(X,Y)=0 ? Because at some values one could influence the other and at others too (but negatively) and so they can cancel each other out?2012-10-26
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    @WuschelbeutelKartoffelhuhn: http://en.wikipedia.org/wiki/Covariance#Uncorrelatedness_and_independence2012-10-26
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    @Alex: Makes sense. thanks2012-10-26

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