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Math Help - Joint density function

  1. #1
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    Joint density function

    Does anyone know how to calulate the joint density function of X and Z

    in this question

    Let X and Y be independent standard normal random variable, and
    Z (equals) aX (plus) (1-a^2)^1/2 * Y
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by frog View Post
    Does anyone know how to calulate the joint density function of X and Z

    in this question

    Let X and Y be independent standard normal random variable, and
    Z (equals) aX (plus) (1-a^2)^1/2 * Y
    Z is a normal, since this is linear in X and Y.
    Thus just find the mean and variance of Z and you're done.
    You can use MGFs, but that silly.
    It does prove what I'm saying though.

    Oh gave you the marginal of Z.
    If you want the joint of X and Z, you need to transform, calc 3 from X,Y to X,Z.
    Obtain the jacobian and substitute from one density to the other.
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