a) Suppose that X and Y are independent. Is it true that their joint CDF satisfies F X,Y(x,y) = F X(x) F Y(y), for all x and y?

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b) Suppose that F X,Y (x,y) = F X (x) F Y(y), for all z and y. Is it true that X and Y are independent?

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Answer 1

a) True. For independent random variables \(X\) and \(Y\), the joint cumulative distribution function (CDF) satisfies the relationship: \[ F_{X,Y}(x,y) = F_X(x) F_Y(y) \] for all \(x\) and \(y\). This is a fundamental property of independent random variables.

b) True. If the joint CDF \( F_{X,Y}(x,y) = F_X(x) F_Y(y) \) holds for all \(x\) and \(y\), this implies that \(X\) and \(Y\) are independent random variables. This is because the factorization of the joint CDF into the product of the marginal CDFs is the definition of independence for random variables.