Given that X and Y are RV supported on [ 2 , <mtext>&#xA0;</mtext> 3 ] , If the

Sattelhofsk 2022-06-26 Answered
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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Misael Li
Answered 2022-06-27 Author has 14 answers
Step 1
Recall Z 1 , Z 2 , are independent if and only if their joint MGF is the product of their individual MGFs:
E [ e t 1 Z 1 + t 2 Z 2 ] = E [ e t 1 Z 1 ] E [ e t 2 Z 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 MGF

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