Roll a die. Let $X$ denote the number on which the die lands on. Toss an unbiased coin $X$ times and let $Y$ denote the number of Heads.
Are $X$ and $Y$ independent or dependent events? If $X = i$ (for $1 \leq i \leq 6)$, then it does not decide the value of $Y$. It does limit the possibilities of $Y$ but does not determine it always.
Since $X$ and $Y$ are random variables, choose two events, say $X=2$ and $Y=0$.
I am trying to figure out how in 2. Independence, Question. 3 whether an event determines the other(s) is associated with independence, is different from the case I wrote in the post here.
Best Answer
Saying $X$ and $Y$ are independent events means that for any $x,y$ we have $\Pr(X=x\text{ and }Y=y)=\Pr(X=x)\Pr(Y=y)$.
It's easy to see that can't be true here. Certainly $\Pr(X=2)>0$, and $\Pr(Y=4)>0$, but $\Pr(X=2\text{ and }Y=4)=0$.
By the same argument (for discrete random variables), if knowing $X$ limits the possibilities for $Y$ in any way then they cannot be independent. It certainly doesn't have to determine $Y$ completely.