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I have known how to tell whether two continuous random variables or two discrete random variables are independent. But is there a genetic method to tell two arbitrary random variables are independent? For example, a continuous and a discrete? Thank a lot for your kind answer.

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    $\begingroup$ Joint CDF is a product of marginal CDF's. $\endgroup$
    – kludg
    Jun 8, 2020 at 2:34
  • $\begingroup$ In some cases characteristic functions provide an easy way of proving independence. $\endgroup$ Jun 8, 2020 at 5:25
  • $\begingroup$ Sometimes a Necessary condition can be useful...Necessary Condition for stochastic independence of 2 rv's is that their joint domain is rectangular so, for example, if their domain is a triangle, you immediately know that the 2 variables are NOT independent $\endgroup$
    – tommik
    Jun 8, 2020 at 6:01
  • $\begingroup$ If CDF is unknown, there are only sample values. Can use hypothesis testing to test the independence? For example, we can use contingency table and chi-square testing to test whether two discrete random variables are independent or not. This's a tutorial courses.lumenlearning.com/introstats1/chapter/… $\endgroup$
    – Jackory
    Jun 8, 2020 at 7:23

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Independence by definition means that for arbitrary RVs $A$, $B$; $P(A)P(B) = P(A \cap B)$. That is the generic method.

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  • $\begingroup$ Your definition of independence makes sense if $A, B$ are events (rather than random variables). $\endgroup$
    – Michael
    Jun 8, 2020 at 5:58

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