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Thread: sum of normal distribution help

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    sum of normal distribution help




    X~N(3,5) Y~N(-7,2)
    What is the distribution of Z=4X-Y/3 ?

    is it just

    =- Y/3 + 4X
    =- Y~N(-7/3,2/3) + X~N(-7/3 + 3,2/3 + 5)
    = X~N(-7/3 + 3,2/3 + 5) - Y~N(-7/3,2/3)

    ?

    What is the covariance of X and Z?

    I don't understand how to do this...

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    Re: sum of normal distribution help

    Again you need several basic properties:

    1. If X, Y are independent normal random variables,
    then Z = X + Y is also normally distributed.

    2. If X is a normal random variable,
    then aX + b is also normally distributed.

    3. E[X + Y] = E[X] + E[Y]

    4. Var[X + Y] = Var[X] + Var[Y] + 2Cov[X, Y]

    5. If X, Y are independent,
    then Cov[X, Y] = 0

    6. Cov[X + Y, Z] = Cov[X, Z] + Cov[Y, Z]

  3. #3
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    oh right, i forgot to mention that X, and Y are independent.

    i'm not understanding how to multiply normal distributions by constants,
    is 4*X~N(3,5) = X~N(12,20) ?

    EDIT: i now get Z~N(43/3, 20.011) is this right?

    and i think i figured out the covariance

    Cov(X,Z)=Coz(X,4X-1/3*Y)
    =Coz(X+0Y,4X-1/3*Y)
    =4Cov(X,X)-1/3 Cov(X,Y)
    =4Var(X)-1/3(0) <--- by independence
    =4Var(X)
    =4*5
    =20

    Is this right?
    Last edited by Dason; 11-21-2010 at 04:39 PM. Reason: Merged 4 posts into 1

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