Given N independent random variables, when does independence stops holding?

Question

I have the following problem: Given $X_1, X_2..X_N$ independent r.v.

$\{X_1,X_4 X_1\}$ are independent of $\{X_3,X_2X_1\}$, but $X_4$ is not independent from $X_4 X_1$

Is that statement true or false? Why?
I believe it has to be False since both sets contain $X_1$ so if I had information on the second one, then the probability of the first would be altered, therefore they are not independent, the same with the second part. I am not entirely sure, and i do not know how to prove it in a formal way. I believe it has something to with the $\sigma$ algebra generated by the r.v., but honestly i dont know.

Thanks a lot in advance!

Answer 1

Why should $\{X_1, X_4 X_1\}$ be independent of $\{X_2, X_2 X_1\}$? In particular, there is no reason for $X_1$ and $X_1 X_2$ to be independent, and it's easy to construct examples where they are dependent. For example, suppose both $X_1$ and $X_2$ have possible values $1$ and $2$. Then $X_1 X_2 = 1$ implies $X_1 = 1$.

Answer 2

$$ P(\{X_1,X_1 X_4, X_3, X_2 X_1\} = (a,b,c,d))=\\ P(X_1 = a) P(X_3 = c) P({X_4, X_2} = (b/a,d/a)|X_1=a,X_3=c)\\ P(X_1 = a) P(X_3 = c) P(X_4 = b/a) P (X_2 = d/a)\\ P(\{X_1, X_1 X_4\}=(a,b)) P(\{X_3, X_1 X_2\}=(c,d)) $$

Therefore, $\{X_1, X_1 X_4\},\{X_3, X_1 X_2\}$ are independent $\forall a$ such that $P(X_2 = d/a) = P(X_1 X_2 = d)$.