Independence of unknown Random variables

Suppose X and Y are two unknown random variables with Pdf's f(x) and g(x) then by looking at their graphs can we say any thing about their independence.

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1 Answer

No, its not possible. Independence of two random variables happens if their joint distribution function can be decomposed into product of marginal distribution functions of $X$ and $Y$. Thus you need to know the Joint Distribution Function. If the joint distribution $F_{XY}(x,y)=F_{X}(x)F_{Y}(y)$ where $F_{X}(x)$ and $F_{Y}(y)$ are marginal distributions of X and Y respectively, then they are independent.

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Thank you. I'll just add that even if the pairwise joint distribution is known for, say, 3 random variables, it doesn't necessarily equal the mutual joint distribution. In other words (loosely), AB, AC, and BC may be independent, yet ABC may not be. – alancalvitti Jan 11 '13 at 6:22
Yes, you are right, thanks for adding that in. – dineshdileep Jan 11 '13 at 6:23