This question is actually based on a previous question that I asked here, with some minor edits:
Given and are independent discrete random variables with
$$\mathbb{E}[]=0, \mathbb{E}[]=1, \mathbb{E}[()^2]=8, \mathbb{E}[()^2]=9, \mathbb{E}[Z]=1, \mathbb{E}[(Z)^2]=9$$
and
$$Var()=Var()=Var(Z)=8$$
Let $=$ and $=Z(+)$.
$A$ and $B$ are independent if the distribution of X is not influenced by the values taken by Y, and vice versa.
So if I understand this correctly, the mean of the values changes when substituting in different values for $X=x$ and $Y=y$ respectively, then they're dependent, else they're independent. Please correct me if I'm wrong.
Same goes for conditional probability distribution of $X$ and $Y$ given ${\displaystyle Z}$, like seen in the defenition below.
Conditional independence of random variables according to wikipedia:
That is, ${\displaystyle X}$ and ${\displaystyle Y}$ are conditionally independent given ${\displaystyle Z}$ if and only if, given any value of ${\displaystyle Z}$, the probability distribution of ${\displaystyle X}$ is the same for all values of ${\displaystyle Y}$ and the probability distribution of ${\displaystyle Y}$ is the same for all values of ${\displaystyle X}$
My question is
Is there a way to tell if $A$ and $B$ are independent by calculating the expectation of $\mathbb{E}[A]$ and $\mathbb{E}[B]$, and also the variance of $Var(A)$ and $Var(B)$ and using those values to determine their independence? If so, how?
The same question applies for conditional independence e.g. using the expecation and variance to determine if and are conditionally independent, by conditioning on either $=\{{0,1,2,...}\}$ or $Y=\{{0,1,2,...}\}$.
After all, like calculated in my other question:
$$(∣=1)=(∣=1)=()=(∣X=1)=(Y∣X=1)=(Y)$$
Since $$Var()=Var()=Var(Z)=8$$
I am really struggling with the topic of independence so I beg you to please bare with me on this one. Thanks!
UPDATE
Cheers to @angryavian for denoting the following
When $A$ and $B$ are not independent, $E[AB]\overset{?}{=}E[A]E[B]$ does not necessarily hold.
Proving that $A$ and $B$ are not independent with
$$[AB]≠[A][B]$$ $$[XY*Z(X+Y)]≠[XY]*[Z(X+Y)]$$
Substituting in the values for both sides yields $8≠0$, thus $A$ and $B$ are dependent.
The only thing I still don't get is the conditional independence.
How can I tell if $$ and $$ are conditionally independent, given =0 or =1?
I thought about substituting in the values e.g. =1 then $A=Y$ and $B=Z(1+Y)$