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Math Help - Show standard normal variables

  1. #1
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    Show standard normal variables

    Hi

    If X and Y has joint density function

    f_{X,Y}(x,y)=\begin{cases} \frac{2|x|+y}{\sqrt{2\pi}}e^{-\frac{(2|x|+y)^2}{2}}, y \geq -|x| \\ 0 , y < -|x| \end{cases}

    ,how do I prove X and Y are standard normal variables?

    Also show that they are not independent, but uncorrelated.


    For the first part, can I calculate the marginal density functions and they would be the density function of a standard normal?

    Or should I use moment generating functions?

    Thanks
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  2. #2
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    Just find the marginals to show that X, Y are standard normals. I did it for X and it really isn't that bad.

    To see that they aren't independent, you just have to look at the support. Compare f_{X|Y} (0|-3) to f_X (0).

    Given that you know EX = EY = 0, all you have to do to show they are uncorrelated is show EXY = 0.
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  3. #3
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    Where did I go wrong?

    f_{X}(x)=\int_{\mathbb{R}}f_{X,Y}(x,y) \, dy = \int_{-|x|}^{\infty} \frac{2|x|+y}{\sqrt{2\pi}}e^{-\frac{(2|x|+y)^{2}}{2}} \, dy =
    =\left[y+2|x|=t\right] = \int_{|x|}^{\infty}te^{-t^2/2} \frac{dt}{\sqrt{2\pi}} = \left[-\frac{2e^{-t^2/2}}{\sqrt{2\pi}}\right]_{|x|}^{\infty}= 2 \cdot \varphi(x)

    , where  \varphi(x) is the density of a standard normal random variable.
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  4. #4
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    \int_{a} ^ b te^{-t^2 / 2} \ dt = -e^{-t^2 / 2}\big| _a ^ b
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  5. #5
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    Ah yes, how careless of me. =)

    For finding f_{Y}(y), Im having a bit of trouble with the integration boundaries.
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  6. #6
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    Draw a picture. Graph the support of f_{XY} as a region on the plane. That should make it easier to figure out the boundaries.

    If y \le 0 then you need to integrate over (-\infty, y) \cup (-y, \infty). Make sure you see why, and figure out what interval to integrate over for y > 0.
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