# How to determine whether two arbitrary random variables are independent?

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.

• Joint CDF is a product of marginal CDF's. Jun 8, 2020 at 2:34
• In some cases characteristic functions provide an easy way of proving independence. Jun 8, 2020 at 5:25
• 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 Jun 8, 2020 at 6:01
• 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/… Jun 8, 2020 at 7:23

Independence by definition means that for arbitrary RVs $$A$$, $$B$$; $$P(A)P(B) = P(A \cap B)$$. That is the generic method.
• Your definition of independence makes sense if $A, B$ are events (rather than random variables). Jun 8, 2020 at 5:58