I am doing a probability course HW and run into trouble with the following problem: Given that X and Y are RV supported on [2,3], If the correlation coefficient of X^t and Y^s is 0 for any s,t ∈ [2,3], then check if X and Y are independent. The homework hint says we may need to consider the moment-generating function to solve this problem, while I have no idea at all. I know that if the correlation coefficient is 0, then E(X^t*Y^s)=E(X^t)E(Y^s), but I do not know how to link this equation with moment generating function and independence. Could anyone give a hand on this?
2026-04-01 05:06:43.1775020003
Given that X and Y are RV supported on [2,3], If the correlation coefficient of X^t and Y^s is 0 for any s,t ∈ [2,3], then X and Y are independent.
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Recall $Z_1,Z_2$ are independent if and only if their joint MGF is the product of their individual MGFs:
$$E[e^{t_1Z_1+t_2Z_2}]=E[e^{t_1Z_1}]E[e^{t_2Z_2}].$$
To show $X,Y$ are independent, it suffices to show $\ln X,\ln Y$ are independent. Can you show $\ln X,\ln Y$ are independent using the uncorrelated condition and MGFs?