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Definition for independent random variables:

The countably valued random variables $X1,...,X_n$ are said to be independent iff for numbers $x_1,...x_n,$ we have $$P(X_1=x_1,...,X_n=x_n)=P(X_1=x_1)...P(X_n=x_n)$$

Deduce further by replacing single values $x_i$ by arbitrary sets $S_i$:

For arbitrary countable sets $S_1,...S_n:$ $$P(X_1\in S_1,...,X_n\in S_n)=P(X_1\in S_1)...P(X_n\in S_n)$$

Definition for independent events:

the n events $A_1,A_2,...,A_n$ are independent if the intersection of any subset of them has its probabilities of the individual events.

For any subset$(i_1,i_2,...,i_k) \text{ of } (1,2,...,n):$ $$P(A_{i_1}∩A_{i_2}∩...∩A_{i_k})=P(A_{i_1})P(A_{i_2})...P(A_{i_n})$$

my question: For indenpent random variables, does the property still apply on any of the subsets? if not provide an example.

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    $\begingroup$ Yes it still holds $\endgroup$ Aug 5, 2020 at 19:14

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For independent random variables, does the property still apply on any of the subsets?

Yes.  Just let $A_i$ be the event that $X_i\in S_i$. Any countable set of such events— ${\{X_i\in S_i\}}_{i=1}^n$ for any collection of arbitrary set of values ${\{S_i\}}_{i=1}^n$ —shall be mutually independent exactly when the corresponding indexed set of random variables– ${\{X_i\}}_{i=1}^n$ –are mutually independent.

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