Given $X_1, X_2..$ independent real r.v., say if $\{X_1 X_2, X_1 X_3, X_1 X_4, \ldots \}$ are always independent

Question

I have a set of problems about independence of RV which I'm having quite a hard time solving. I think I know the theory, but when it's time to get the hands dirty, I do not know where to begin.
Here is one of them

Given $X_1, X_2,\ldots$ independent real r.v., say if the following proposition is always true or not $\{X_1 X_2, X_1 X_3, X_1 X_4, \ldots\}$ are independent.
I know that my objective is to prove that
$$ P(\{X_1 X_2, X_1 X_3, X_1 X_4, \ldots\}) = P(X_1 X_2) \cdot P(X_1 X_2) \cdot P(X_1 X_3)\cdots $$ As the definition of independence. But I do not find the way.
What I tried so far:\

1. Conditioning on $X_1$ like $P(a X_2\mid X_1=a, aX_3\mid X_1=a,\ldots) P(X_1=a)$, given the $X_is$ are independent that factorizes, but i cannot put the "times $X_1$" back into each probability.\
2. Thinking as a cartesian product $P(\{X_1\} \times \{X_2,X_3 \dots \})$, and I couldn't actually elaborate much here.\

I am looking to both: A hint to advance with this one, and maybe some insight on how to address these kind of problems, since I have a few left.

Thanks in advance

Answer 1

Generally for problems like that, you want to construct a(n) (counter-)example and play around with it.

As a hint, here is a particular random variable to think about: Let $X_1$ be Bernoulli(1/2), that is, it is $0$ with probability 1/2, and $1$ with probability 1/2. Now, construct some $X_i$s for yourself, and then ask questions like "What is the probability that $X_1 X_i = 0$ given $X_1 X_j = 0?$"

As a general problem solving strategy, you should make sure to not have typos in your problem statements (as was outlined in the comment section), because that could be a result of misunderstanding some fundamental concepts. Otherwise, if it is a proof question, try to play with simple examples: it is easier to see why something should be true by looking at smaller examples, and eventually, you will either see a pattern or you will come up with a counter-example.

Answer 2

This is obviously untrue. Consider the case of $X_2 = X_3 = X_4 = ... = 1$ and $X_1 = N(0,1)$. Then $X_1, X_2, ...$ are jointly independent, but the products are not.