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Covariance of two dependent variables

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data

    See figure attached

    2. Relevant equations



    3. 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!
     

    Attached Files:

  2. jcsd
  3. Mar 12, 2013 #2
    Here is the last page of my attempt.
     

    Attached Files:

  4. Mar 12, 2013 #3

    haruspex

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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?
     
  5. Mar 12, 2013 #4
    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)?
     
  6. Mar 12, 2013 #5

    haruspex

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    Quite so. E[Y] = E[|X|+N] = E[|X|]+E[N].
     
  7. Mar 12, 2013 #6
    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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