TITLE: Two random variables and their sum
QUESTION [10 upvotes]: $X$ and $Y$ are random variables uniformly distributed on $[a,b]$. Could the random variable $Z=X+Y$ be uniformly distributed if $X$ and $Y$ are dependent and correlation between them doesn't equal $1$ or $-1$?

REPLY [4 votes]: We produce a discrete example, because the details are more pleasant to verify. The same idea leads to a continuous example. 
Let $X=-3,-1,1,3$ each with probability $1/4$. Suppose that when $X=-3,-1,1,3$ respectively, then $Y=1,3,-3,-1$ respectively. Then $X+Y$ takes on the values $-2$ and $2$, each with probability $1/2$.
Since $X$ and $Y$ have mean $0$, their covariance is $E(XY)$ is $-3$. The variance of $X$ and of $Y$ are each $5$. So the correlation coefficient of $X$ and $Y$ is $-\frac{3}{5}$. 
Continuous examples can be made using a similar strategy.