Covariance of two dependent variables

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Homework Help Overview

The discussion revolves around the covariance of two dependent variables, X and Y, with a focus on understanding their relationship and the calculations involved in parts (b) and (c) of the problem. The original poster expresses uncertainty about their approach and results in these sections.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to clarify their understanding of the dependency between X and Y, questioning their calculations in parts (b) and (c). Some participants discuss the expected covariance and the nature of the relationship between the variables, while others raise questions about specific calculations and assumptions made in the problem.

Discussion Status

Participants have provided feedback on the original poster's attempts, noting that part (b) appears correct while questioning the reasoning in part (c). There is an ongoing exploration of the calculations related to the expected values and covariance, with some guidance offered on how to approach the calculations.

Contextual Notes

There is mention of a figure that is referenced but not included in the discussion. Additionally, the original poster expresses confusion regarding the nature of the expected value of Y and the implications of the relationship between X and Y, indicating potential gaps in information or assumptions that need to be clarified.

jegues
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Homework Statement



See figure attached

Homework Equations





The Attempt at a Solution



I am not concerned with part (a), I have deduced that indeed X and Y are dependent.

I'm not sure if I have done part (b) correctly, and I am quite certain I have done part (c) incorrectly, but I couldn't think of what else to do.

Am I on the right track in parts (b) and (c)? What is incorrect, or how should I approach the problem?

Thanks again!
 

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Here is the last page of my attempt.
 

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B) looks ok. you should expect zero covariance. The two are clearly anticorrelated where X is negative and, equally, correlated where X is positive.
In c), something strange at the bottom of the first sheet. How did μXμY become X?
 
haruspex said:
B) looks ok. you should expect zero covariance. The two are clearly anticorrelated where X is negative and, equally, correlated where X is positive.
In c), something strange at the bottom of the first sheet. How did μXμY become X?

I calculated μx for that case, it was found to be 1.

Since Y = |X| + N, and X is in the range of (0,2) |X| = X.

Thus, Y = X + N, I figured this pdf would simply be the pdf of N (it is known to be Gaussian) shifted to the right by X, having a mean value of X. Hence, μy = X.

I don't think that is correct, because I think μy should be a number.

Where did I go wrong? How do I fix part (c)?
 
jegues said:
I don't think that is correct, because I think μy should be a number.
Quite so. E[Y] = E[|X|+N] = E[|X|]+E[N].
 
haruspex said:
Quite so. E[Y] = E[|X|+N] = E[|X|]+E[N].

Okay so from this,

[tex]E[Y] = E[X] + 0 = \mu_{X} = 1 = \mu_{Y}[/tex]

but I am still stuck in finding,

[tex]\sigma_{XY} = E[(X-\mu_{X})(Y-\mu_{Y})] = E[(X-1)(Y-1)] = E[(X-1)(|X|+N-1)][/tex]

Any ideas?
 

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