Independence of Bernoulli random variables.

Question

there. Here is a question for you. Let $p,q \in [0,1]$, and let $X \sim Be(p)$, $Y \sim Be(q)$. Can we conclude that random variables $X$ and $Y$ are independent if $p \neq q$? Thanks :)

Answer 1

No. Independence has nothing to do with the parameters.

Answer 2

Counterexample, with $p<q$ without loss of generality: define $Z\sim\operatorname{Be}(p/q)$ independent of $Y$, then take $X=YZ$.