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#1




Independence of x and y (joint)
Hi guys,
Apologies if this may just be a simple question, but I prefer thinking out loud, so here's my question: For the independence of x and y, do the following conditions have to be BOTH met? 1. The probability space is rectangular 2. f(x,y) = f(x) * f(y) I'm quite confused. Thanks in advance! 
#2




The rectangle can be unbounded, but yes. If the support is not rectangular, then the possible values of one variable depend on the value of the other. This is why. If both have a marginal pdf or pmf, then the product of these is their joint pdf or pmf if and only if they are independent.

#4




To be clear, "rectangular" in this case includes the degenerate rectangles: the point and the line.
And the rectangles in question must have edges parallel to the axes. No crooked rectangles. 
#5




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You can easily prove that your (2) implies (1) here which is why (2) is equivalent to independence. 
#6




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