I am interested in computing $Corr(X,Y)$ for $0 \leq x \leq 1$ and $0 \leq y \leq 1$, given two cases: firstly, $F_{X,Y}(x,y)=min(x,y)$ and secondly, $F_{X,Y}(x,y)=xy$.
How would I go about doing this? In the earlier part to this question, I have found that $X$ and $Y$ both follow the standard uniform distribution. I immediately thought of applying the correlation formula, as I can find all the expectations that I need, except for $E(XY)$ - how do I find that?
A worthy note is that this question is worth 6 points - 3 points for each case, so I personally do not think that my professor intended for us to use the correlation formula, as that would require calculations of $E(XY)$, $E(X)$, $E(Y)$, $E(X^2$), $E(Y^2)$, $E(X)^2$ and $E(Y)^2$. I feel like there is a more (un)intuitive approach to this question, which I clearly cannot see.
The first one is simply the distribution fu nction 0f $(X,X)$ where $X\sim unif(0,1)$ and the second one is the distribution function of $(X,Y)$ where $X$ and $Y$ are i.i.d with $unif(0,1)$ distribution. So the covariance is $0$ in second case. In the first case it is $EX^{2}-(EX)^{2}=\frac 1 3 -\frac 1 4=\frac 1 {12}$.