My question is conceptual.
Suppose I have the marginal distribution for two random variables $X$ and $Y$, and find that $E(X)=1.5$ and $E(Y)=3$. Suppose also that these random variables are determined by an integer $N$ which is chosen from 1 to 10 uniformly at random.
e.g., $X = N+c$ and $Y= N/g$ where $c$ and $g$ are some real constant.
Given I have the marginal distributions for $X$ and $Y$, is there a way for me to find $E(XY)$, so that I can check if $X$ and $Y$ are independent? i.e., to check if: $$E(XY)=E(X)E(Y) $$ or $$VAR(X+Y)= VAR(X)+VAR(Y) $$
If so, how would I go about constructing the joint density function? If not, how do I find the independence?
$X=N+c=gY+c$ If $X$ and $Y$ are independent then $X$ would be independent of itself which makes it a constant r.v. Hence, unless $X$ and $Y$ are constant there is no scope for $X$ and $Y$ to be independent.
$EXY=E(gY+c)Y=gEY^{2}+cEY$. You will need $EY^{2}$ to find $EXY$.
In general, even if you know the exact distrbutions of $X$ and $Y$ (not just their means) you cannot determine if they are independent. You will need a knowledge of their joint distribution.