Say $X$ and $Y$ are two random variables where $X\in [-\alpha,\alpha]$, $Y\in [-\alpha,\alpha]$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily identically distributed) that force $Z$ to be uniformly distributed (i.e. $Z\sim \mathcal{U}[-2\alpha,2\alpha]$)?
As the sum of $N$ random variables with zero mean resembles Gaussian distribution with zero mean, I suspect it is not possible to find two such random variables. Do you know any counterexample?
Best Answer
Here is another one.
Recall if $U_n$ are IID with Bernoulli distribution $$ \mathbb P[U_n=0]=\frac{1}{2},\qquad \mathbb P[U_n=1]=\frac{1}{2}, $$ then $$Z = \sum_{n=1}^\infty U_n 2^{-n} $$ is uniformly distributed on $[0,1]$. So let $$ X = \sum_{n\text{ even}} U_n 2^{-n},\qquad Y = \sum_{n\text{ odd}} U_n 2^{-n} $$ to get independent $X,Y$ with $X+Y=Z$.