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Thread: Poisson distribution and covariance

  1. #1
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    Poisson distribution and covariance

    Hello ! I have trouble getting to the end of this exercise :

    Y1 et Y2 are two random variables following Poisson distributions with parameters y1 et y2.
    U = Y1 + Y2 et V = aY1 - aY2 with a ≠ 0

    Give the value of the covariance of U and V

    I started with Cov (U,V)= E(UV) - E(U)E(V) = E((Y1 + Y2)a(Y1-Y2)) - E(Y1+Y2) aE(Y1-Y2))

    I know the end result to be obtained is cov(U,V)=a(var(Y1)-var(Y2)). However I don't see how to proceed to get to this form, I tried to develop the expression but ended up with nonsense.


    Thank you for your help !
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  2. #2
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    Re: Poisson distribution and covariance

    are Y1 and Y2 independent?
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  3. #3
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    Re: Poisson distribution and covariance

    I found the problem can be solved using the bilinearity property :

    Cov(Y1+Y2,aY1 - aY2) =aCov(Y1+Y2,Y1-Y2) = aCov(Y1,Y1) - aCov(Y1,Y2) + aCov(Y1, Y2) + aCov(Y2, Y2) = aCov(Y1,Y1) - aCov(Y2,Y2)= a(var(Y1)-var(Y2))

    However I still couldn't work it out with the E[X] formula.


    I think we dont know if Y1 and Y2 are indépendant ? And if cov were equal to 0 with Y1 or Y2 = 0 it doesn't prove independence.
    But U and V are composed of the same variables Y1 and Y2, can we say they are dependent ?
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  4. #4
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    Re: Poisson distribution and covariance

    Hey Jo37.

    For dependency in this situation try setting up the matrix composed of U and V and show that the determinant is non-zero to prove they have independence.
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  5. #5
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    Re: Poisson distribution and covariance

    Like an ad -bc determinant ?
    I don't see how to set up U and V in a matrix.

    Y1 Y2
    aY1 aY2
    ?

    ad -bc = Y1aY2 -aY1Y2 = 0 ? (no idea what I'm doing ahah)
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  6. #6
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    Re: Poisson distribution and covariance

    The coefficients of Y1 and Y2 are in the matrix - not the actual variables.

    The determinant is - 1*a - 1*a = -2a which means they are linearly dependent iff a = 0.

    If a != 0 then they are independent and you can use that assumption.
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  7. #7
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    Re: Poisson distribution and covariance

    Thank you and sorry for the late reply.
    I do not understand how you form the matrix and it doesn't appear in your answer =/. Also where does that determinant formula comes from ?

    Thank you for the help !!
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