If X2 is dependent on X1, how to generate the random sample of (X1,X2)? One scenario is that we know the prior distribution of X1 and functional relationship between X1 and X2, how to generate the sample of (X1,x2) in this scenario.
Another scenario is that we know the correlation matrix of X1 and X2, how to generate the sample of (X1, X2)?
In the first case, you can just execute the functional relationship, e.g. $X_2=X_1^2$ and have your sample, after you successfully sampled $X_1$. Having the correlation for sampling the tuple is not enough. You need to know the joint distribution. Of course you can generate a correlated sample, and treat as if it is your sample, but that wouldn't be associated with a unique joint relationship. For instance, if your correlation is, let's say, $1$, executing the relation $X_2=aX_1$ generates you a $X_2$, that has correlation $1$ with $X_1$ (assuming $a>0$). So, having $X_2=3X_1, X_2=X_1, X_2=0.5X_1$ all generate the same correlation. The situation is not different for $\rho\neq 1$, this was just the simplest example I could think of directly. You need to have information about the joint distribution to create tuple samples.
