Sattelhofsk

2022-06-26

Given that X and Y are RV supported on $[2,\text{}3]$ , If the correlation coefficient of ${X}^{t}$ and ${Y}^{s}$ is 0 for any $s,\text{}t\text{}\in \text{}[2,\text{}3]$ , then X and Y are independent.

Misael Li

Beginner2022-06-27Added 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 $\mathrm{ln}X,\mathrm{ln}Y$ are independent. Can you show $\mathrm{ln}X,\mathrm{ln}Y$ , are independent using the uncorrelated condition and MGF

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 $\mathrm{ln}X,\mathrm{ln}Y$ are independent. Can you show $\mathrm{ln}X,\mathrm{ln}Y$ , are independent using the uncorrelated condition and MGF

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A) R is reflexive

B) R is symmetric

C) R is not anti-symmetric

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Correlation bound

Let x and y be two random variables such that:

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My questions are:

Is there a way where I can bound the correlation between x, y' i.e. Corr(x,y')?I mentioned y' in light of random perturbation, I would like to know what if I don't have that information, where I only know that y' is a estimation of y. Are there any literature that cover it?What is the benefit of OU vs regression for modeling data, say data in the form of ($x,y$) pairs?

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