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Math Help - Probability question normal random variables

  1. #1
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    Probability question normal random variables

    Let Z1 be N(m₁, σ) and Z2 be N(m₂, σ) Gaussian random variables (not
    necessarily independent).

    (a) Prove that X1 = Z1 + Z2, X2 = Z1 − Z2 are independent Gaussian
    random variables.
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  2. #2
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    Re: Probability question normal random variables

    Hint: See if you can compute the covariance of X1 and X2.
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    Re: Probability question normal random variables

    I thought that independant variables have zero covariance but zero covariance does not imply independance. In which case is it sufficient to show that cov(x1,X2)=0?
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    Re: Probability question normal random variables

    For normally distributed random variables, zero covariance implies independence.
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    Re: Probability question normal random variables

    ok thanks I didn't know that was the case.
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    Re: Probability question normal random variables

    I am not sure how to approach part b of this question.
    b)if cov(Z₁,Z₂)=ρσ
    Find the M.G.F of (X₁,X₂)=(2Z₁-Z₂,Z₁+Z₂)
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  7. #7
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    Re: Probability question normal random variables

    I think you are supposed to assume that Z1 and Z2 have a joint bivariate normal distribution. Under that assumption, 2Z1-Z2 and Z1+Z2 have a joint bivariate normal distribution. I think you can compute the means, standard deviations, and correlation of 2Z1-Z2 and Z1+Z2 from the information given. Then you should be able to find the MGF from a known formula for the MGF of a bivariate normal.
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  8. #8
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    Re: Probability question normal random variables

    Thanks i figured it out. It was basically what you stated.
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