Inversion and correlation of Binomial random variable Assume $A$ is $Bin(1, 1/2)$ and $B$ is $Bin(1, 1/4)$, where $X=A+B$ and $Y= AB$ and I am interested in knowing how to find their inversion so that I could use the good old transformation theorem
$$f_{XY} (\cdot, \cdot ) |\mathbf{J}|$$
where $\mathbf{J}$ is the usual jacobian to find their joint distribution. But the inversion seems a difficult task; it seems $A$ and $B$ cannot be made the subject of $X=A+B$ and $Y= AB,$ for example $A = X-B$ then $Y = (X-B)B$, $B$ is a square making it hard. Plugging $A = Y/B$ into $X = (Y/B) + B$, again a quadratic.
Granted that the inversion is possible so that the joint distribution can be found, what is the smarter way for computing the correlation between $X$ and $Y$? Generally, we need to find the Cov$(X,Y)$ and divide by their product of the standard deviations (square root of variances). This approach is always tedious and time consuming.

This approach (inverting the transformation, finding the Jacobian, etc.) is applicable when $A$ and $B$ are jointly continuous random variables, and $f_{A,B}$ is their joint density function. In this problem they are discrete case, so we would do something like
$$ f_{X,Y}(x,y) = \sum_{(a,b)\, s.t. \\ x=a+b,\\ y=ab} f_{A,B}(a,b)  $$
where the sum is over all pairs of values $(a,b)$ that give you the desired $(x,y)$. For example, if $X=1$ and $Y=0$ you need to take $(a,b)=(0,1)$ and $(a,b)=(1,0)$.
Also, you talk about computing the correlation between $A$ and $B$, but in your setup these appear to be independent. Do you mean the correlation between $X$ and $Y$?
