I have two distributions $A$ and $B$ that are i.i.d.
I want to create two distributions $A'$ and $B'$, that have the "same distribution" as $A$ and $B$ (meaning the same probability distribution function), but are correlated (pearson correlation) with correlation $\rho$.
My first approach was to set $A' = B' = (A+B)/2$, but this works only for perfect correlation (or $\rho = -1$). Furthermore the distribution of $A'$ is not neccessarily equal to the distribution of $A$.
If the general case is to complicated, the problem I work on actually only requires $A$ to be uniform and not any given distribution.
This trick works for any $\rho\in[0,1]$: Flip a biased coin whose heads probability is $\rho$. If heads comes up, let $(A',B') = (A,A)$; otherwise let $(A',B') = (A,B)$.
If you want a negative $\rho$, and if $A$ and $B$ are to be uniform on $[0,1]$, flip a coin with heads probability $|\rho|$ and let $(A',B') = (1-A,A)$ if heads comes up and $(A',B') = (A,B)$ otherwise.