Let $(X_1,X_2)$ be a pair of uniformly distributed random variables with $S=[-1,1]\times[0,1]$. Are $X_1$ and $X_2$ independent? Are $X_1$ and $X_2$ correlated.
My approach for independence:
For independence I think what was has to be shown is that $$f(x_1,x_2)dx_1dx_2=f(x_1)dx_1f(x_2)dx_2$$ where for $X_1$ the density is $$f(x_1)dx_1=\frac{1}{1-(-1)}dx_1=\frac{1}{2}dx_1$$ and for $X_2$ the density is $$f(x_2)dx_2\frac{1}{1-0}dx_2=dx_2.$$
I'm very confused on what $f(x_1,x_2)$ is.
My approach for correlation:
No approach yet. But I found out that two random variables are uncorrelated iff $cov[X_1, X_2]=0$. How can I show that for the above problem?
As far as I understand your question we have $f_{X_1, \, X_2}(x_1, x_2) = \mathbf{1}_{[-1, 1] \times [0, 1]}(x_1, x_2) \, \cdot \frac{1}{2}$. We can now get $f_{X_1}$ and $f_{X_2}$ up to a constant by integrating over the other random variable. $$f_{X_1} = \int_\mathbb{R} f_{X_1, X_2}(x_1, x_2) \, dx_2 = \int_\mathbb{R} \mathbf{1}_{[-1, 1] \times [0, 1]}(x_1, x_2) \, \cdot \frac{1}{2} \, dx_2 = \mathbf{1}_{[-1, 1]} \cdot \frac{1}{2} \int_0^1 \, dx_2 = \mathbf{1}_{[-1, 1]} \cdot \frac{1}{2}$$ You can do the same with $X_2$ and use your right idea about the splitting up of the density functions. Regarding correlation check again how independence and correlation are connected and how your result about independence might help you here already.