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Math Help - Show that X+Y is normal

  1. #1
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    Show that X+Y is normal

    If X and Y have a bivariate normal distribution with correlation p, show that X +Y is normally distributed


    This seems like a pretty standard proof but cant find it anywhere. Simply adding the marginal distributions, X+Y is what I tried but how is the correlation coefficient introduced?
    I then looked at convolution but that didnt seem to go the right way either...
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  2. #2
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    Quote Originally Posted by Gekko View Post
    If X and Y have a bivariate normal distribution with correlation p, show that X +Y is normally distributed


    This seems like a pretty standard proof but cant find it anywhere. Simply adding the marginal distributions, X+Y is what I tried but how is the correlation coefficient introduced?
    I then looked at convolution but that didnt seem to go the right way either...
    Use the change of variable theorem:

    Make the transformation U = X + Y, V = X - Y. The inverse transformation is X = \frac{1}{2} (U + V), Y = \frac{1}{2} (U - V). Then |J| = \frac{1}{2}.

    Get the joint pdf of U and V. Integrate wrt to v to get the marginal, which is the required pdf.
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  3. #3
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    By definition of a multivariate normal distribution, if a random vector follows it, say (X,Y), any linear combination of its components follows a normal distribution, X+Y for example.
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  4. #4
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    Thanks for your reply.

    How do we handle the standard deviations of x and y when converting to U and V? Additionally, taking this approach means we start from the marginal distributions of X and Y. Isnt there a way to start from the joint bivariate distribution of X and Y since this includes the correlation term?
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  5. #5
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    Unfortunately this approach doesnt seem to work. I end up with a very messy exponential which doesnt allow separation for the marginal calculation

    Any thoughts? Is this not a standard proof?
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