I know and can even prove that two continuous random variables X and Y are independent iff their joint density function f(x,y) can be written as f(x,y) = h(x)g(y).
Maybe I should carefully look at my proof of this but I do not see why the following two RV are not independent by this theorem.
f(x,y) = 4/9 if 0<= x < y < 3-x and 0 otherwise.
I am thinking of 4/9 as (4/9)*1 = f(x)g(y). That is, f(x) = 4/9 and g(y)=1.
So I conclude that X and Y are independent but this is wrong since for example f(1,2) = 4/9 while f_x(1) = 4/9 and f_y(2) = 4/9.
What am I missing in this theorem??????????