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Thread: gamma distribution 2

  1. #1
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    gamma distribution 2

    I had a similar problem and I was trying to solve this question based on reply (thank you so much!!) but im having a difficult time to do following question:

    For Z distributed on R^3 with Z1, Z2, Z3 IID Z ~ exp(1)

    If i have W = (Z1 ^2, Z2 ^2, Z1Z2) then how can I find expected value and variance?

    Do I still use gamma distribution? I am so confused....
    plz help me!!!
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  2. #2
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    W is the joint density of three i.i.d variates. What does this mean?
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  3. #3
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    []EDIT[]Corrected post. Please refer to Matheagle and Moo's expert posts below.
    Last edited by Anonymous1; Mar 17th 2010 at 08:25 AM.
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    MHF Contributor matheagle's Avatar
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    are you asking for the mean and covariance matrix of a vector?
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  5. #5
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    Thank you so much !!!!

    Jacobian tranfomation? I am not familiar with it T-T

    I need to find E(W) and Var(W)........that's what question was asking does it help?
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  6. #6
    Super Member Anonymous1's Avatar
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    I think you can use my method. Unless there are any objections?

    Let me know if you get stuck.
    Anonymous
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  7. #7
    MHF Contributor matheagle's Avatar
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    $\displaystyle E(W) = \left(E(Z_1 ^2), E(Z_2 ^2), E(Z_1)E(Z_2)\right)=(2,2,1)$

    The variance/covariance matrix has in the $\displaystyle ij^{th}$ position the covariance between Zi and Zj.
    Note that the covariance between Z1 and Z1 is its variance.
    It's a 3 by 3 symmetric matrix, where you have the 3 variances down the diagonal and the covariances in the other positions.
    Last edited by matheagle; Mar 17th 2010 at 12:15 AM.
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  8. #8
    Moo
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    Quote Originally Posted by Anonymous1 View Post
    Since they are i.i.d you can multiple the three densities $\displaystyle f_{Z}^2(z)$ where $\displaystyle f_{Z}(z)$ is the density of an $\displaystyle exponential(1)$ to find your joint distribution. See what you get. Does it look gamma? Then simply compute the first and second moments to find the variance of W.

    $\displaystyle \int_0^{\infty} z\times f_{Z_1}^2(z)*f_{Z_2}^2(z)*f_{Z_3}^2(z)dz$

    $\displaystyle \int_0^{\infty} z^2\times f_{Z_1}^2(z)*f_{Z_2}^2(z)*f_{Z_3}^2(z)dz$

    wait a minute... You may have to use a Jacobian transform. Does anyone know?
    What the... ?

    The pdf of $\displaystyle Z^2$ when Z has the pdf $\displaystyle f_Z$ is certainly not $\displaystyle f_Z^2$
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  9. #9
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    Thank you so much!!! so you mean that 3 by 3 symmetric matrix will look like

    var (Z1^2) cov (Z1^2, Z2^2) cov (Z1^2, Z1Z2^2)
    cov (Z2^2, Z1^2) var (Z2^2) cov (Z2^2, Z1Z2^2)
    cov (Z1Z2^2, Z1^2) cov (Z1Z2^2, Z2^2) var (Z1Z2^2)

    but aren't they all 1s for var(Z1^2) and var(Z2^2) and even var(Z1Z2^2)??? in this case??? im not sure....T-T
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  10. #10
    Moo
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    Quote Originally Posted by dymin3 View Post
    Thank you so much!!! so you mean that 3 by 3 symmetric matrix will look like

    var (Z1^2) cov (Z1^2, Z2^2) cov (Z1^2, Z1Z2)
    cov (Z2^2, Z1^2) var (Z2^2) cov (Z2^2, Z1Z2)
    cov (Z1Z2, Z1^2) cov (Z1Z2, Z2^2) var (Z1Z2)

    but aren't they all 1s for var(Z1^2) and var(Z2^2) and even var(Z1Z2^2)??? in this case??? im not sure....T-T
    False

    Var(Z^2)=E(Z^4)-[E(Z^2)]^2

    And it's easy, the moments of an exp(1) are : E[X^k]=k!

    So Var(Z^2)=4!-2^2=24-4=20

    For the covariance, we have for example cov(Z2^2,Z1^2)=0, because Z1 and Z2 are independent.


    And you messed up some things : you're dealing with Z1Z2, not Z1Z2^2. I corrected it.
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  11. #11
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    Wow, thank you so much. I have one last question

    Now i have Var(Z1Z2) = E[(Z1Z2)^2] - [E(Z1Z2)]^2 = ? - 1^2
    How can i find E[(Z1Z2)^2]? I can't use the moments of an exp(1) for this, can I?
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  12. #12
    Moo
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    ... It's algebra : (Z1Z2)^2=Z1^2 Z2^2
    And since Z1 and Z2 are independent, we'll have E[(Z1Z2)^2]=E[Z1^2]E[Z2^2]
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  13. #13
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    Quote Originally Posted by Moo View Post
    ... It's algebra : (Z1Z2)^2=Z1^2 Z2^2
    And since Z1 and Z2 are independent, we'll have E[(Z1Z2)^2]=E[Z1^2]E[Z2^2]
    oops.... i kept forgetting about the meaning of the independent
    Thanks!!

    Now I have EW = (2,2,1) and VarW = (20, 20, 3)
    Problem solved!!! YEY!!
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