Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.

To be a bit more precise:

Suppose we know $X$ and $Y$ are iid and

$$ X+Y \sim UNIF({1, \dots , n})$$

Does this necessarily imply that both $X$ and $Y$ are discrete uniform as well?

Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.

To be a bit more precise:

Suppose we know $X$ and $Y$ are independent and

$$ X+Y \sim UNIF({1, \dots , n})$$

Does this necessarily imply that both $X$ and $Y$ are discrete uniform as well?

Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.

Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.

To be a bit more precise:

Suppose we know $X$ and $Y$ are iid and

$$ X+Y \sim UNIF({1, \dots , n})$$

Does this necessarily imply that both $X$ and $Y$ are discrete uniform as well?