Do there exist two such independent variables on $\Omega$?

Question

Let's s suppose the space of outcomes $\Omega$ consists of 5 elements, each of the outcomes has a positive probability. Do there exist two independent variables on $\Omega$ such that both of them can take only 5 different values.

I don't know how to approach this problem. Any help would be appreciated.

Answer 1

If $X$ and $Y$ can take on at most five different values, then $X$ and $Y$ constant works, since constant variables are always independent. In general, note that if $X$ and $Y$ are random variables on $\Omega=\{1,2,3,4,5\}$ taking on exactly five different values, then $X$ and $Y$are injective, implying that

$\mathbb{P}(X= X(1), Y=Y(2))=\mathbb{P}(\emptyset)=0,$

while

$$ \mathbb{P}(X=X(1))\mathbb{P}(Y=Y(2))=\mathbb{P}(1)\mathbb{P}(2)\neq 0 $$ by assumption. Thus, $X$ and $Y$are not independent.