Independence betweeen sets of random variables

Question

Given two random variables $A$ and $B$, I know we can call them independent if their joint PDF is factorizable to a product of their marginal PDFs:

$f_{A,B}\left(A,B\right)=f_{A}\left(A\right)\cdot f_{B}\left(B\right)$

My problem: I have two sets of random variables, $\mathcal{S}_1=\left\{A,B \right\}$ and $\mathcal{S}_2=\left\{C,D \right\}$. I'd like to be able to show that the first set of random variables is independent of the second. I don't care about the relationship between $A$ and $B$, and the relationship between $C$ and $D$. Intuitively, I would understand "independence" here to mean that, given a single realization of all four random variables, the value of the realization for the first two will give me no additional help in predicting the value of the realization for the second two. Strictly speaking, it seems to me that this is equivalent to showing that the four-dimensional joint PDF factorizes to a product of the two two-dimensional PDFs:

$f_{A,B,C,D}\left(A,B,C,D\right)=f_{A,B}\left(A,B\right)\cdot f_{C,D}\left(C,D\right)$

Is my reasoning correct?

Answer 1

Yes, it is correct (modulo a few typographical mistakes) provided you get away from having the arguments of the PDFs be the same as the subscripts. The arguments are real variables; the subscripts are random variables and no, they don't even need to have the same letters in different cases (upper vs lower) or fonts. What you need to write for complete correctness is $$f_{A,B}\left(x,y\right)=f_{A}\left(x\right)\cdot f_{B}\left(y\right), \quad -\infty < x, y < \infty$$ and $$f_{A,B,C,D}\left(w,x,y,z\right) =f_{A,B}\left(w,x\right)\cdot f_{C,D}\left(y,z\right), \quad -\infty < w, x, y, z < \infty$$ where those comments following the equalities are very important. The joint density must be expressible as the product of the marginal densities at every point in $\mathcal R^2$ and $\mathcal R^4$ respectively.