Is it possible to generate a uniform distribution by summing variables drawn other distributions? If I have two processees $x_{1}$ and $x_{2}$, what distribution(s) would they need to have in order for $x_{1} + x_{2}$ to be uniformly distributed?
 A: The answer is yes.
Dependent $X_1$ and $X_2$ example:
If $X_1$ and $X_2$ need not be independent, then it's trivial.
Let $X_1$ be a random variable following any distribution. Let $Z$ be a random variable following the uniform distribution (eg. from 0 to 1). Compute the distribution of $X_2 = Z - X_1$. Then $X_1 + X_2$ will be uniform.
Independent $X_1$ and $X_2$ example:
As the answer by @grand_chat points out, this answer on math.stackexchange provides a clever example.
A: If you are looking for two random variables whose sum is uniform over an interval, here are some pointers in the independent case.
If $X$ and $Y$ are independent random variables, and ...


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*... $X$ and $Y$ both have densities, then the answer is no. This can be seen from the density of $X+Y$, which will be the convolution of the densities of $X$ and $Y$. The convolution $f_{X+Y}(x)$ will peak when the densities of $X$ and $Y$ are 'aligned', and will decrease from that peak.

*... $X$ and $Y$ have the same distribution, then the answer is no. One proof involves characteristic functions and showing that the CF of $X$ is impossible.
If $X$ and $Y$ are independent but not identically distributed, the answer is yes, it is possible. By the above, at least one of $X$ and $Y$ cannot have a density. Examples:


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*If $Y$ is constant, there's an example in Zahava Kor's answer: Let $X$ have uniform distribution.

*A more bizarre example is seen here.
A: First - X1 and X2 are not processes but random variables. Second - you did not specify whether they have to be independent ot not. If Independent - X+c will work where X is Uniform and C is a constant. If not necessarily independent - then X1=X2=X/2 where X is Uniform will work, because X/2 + X/2 = X and X is Uniform.
