# The variance of a sum of uniforms

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?

• 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. – Alex Oct 26 '12 at 3:22
• 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. – joriki Oct 26 '12 at 4:15
• @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? – Wuschelbeutel Kartoffelhuhn Oct 26 '12 at 5:10
• @WuschelbeutelKartoffelhuhn: en.wikipedia.org/wiki/… – Alex Oct 26 '12 at 5:20
• @Alex: Makes sense. thanks – Wuschelbeutel Kartoffelhuhn Oct 26 '12 at 5:28

## 1 Answer

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.