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Math Help - Transforming the Normal distribution

  1. #1
    Senior Member chella182's Avatar
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    Transforming the Normal distribution

    X is a standard normal random variable. Show that the PDF of Z=(X+1)^2 is

    (8\pi z)^{-\frac{1}{2}}e^{-\frac{1}{2}(z+1)}\left(e^{\sqrt{z}}+e^{-\sqrt{z}}\right)

    I've tried this loads of times but I can't get anywhere near the answer
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  2. #2
    MHF Contributor matheagle's Avatar
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    I've already shown you the technique.
    Why don't you post what you've done and we can find what's wrong.
    There's a basic formula in most books for 2-1 transforms, so all you have to do is plug into that.
    I'm sure you can find it online, but it's easy to do it from scratch.
    Last edited by matheagle; November 26th 2009 at 06:36 PM.
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  3. #3
    Senior Member chella182's Avatar
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    I know you have, and I tried applying that (along with using a similar example we had in our notes) but didn't get anywhere near the required answer (that I'm aware of). I'll just type exactly what I've written.

    F_{Z}(z)=P(Z\leq z)
    =P(X^2+1\leq z)
    =P(X+1\leq\sqrt{z})
    =P(X\leq\sqrt{z}-1)
    =\Phi(\sqrt{z}-1)

    That's practically identical to what we wrote for the example in our notes. Then we wrote

    \frac{d}{dz}(\Phi(\sqrt{z}-1)
    =\frac{1}{2}z^{-1/2}\Phi'(\sqrt{z}-1)

    Then I've got that \Phi'(z)=\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2} so

    f_{Z}(z)=\frac{1}{2}z^{-1/2}\frac{1}{\sqrt{2\pi}}e^{-(\sqrt{z}-1)^2}

    And then this is where I start going off miles away from the answer.
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  4. #4
    MHF Contributor matheagle's Avatar
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    no, look at your start

    x^2+1\ne (x+1)^2

    But that seems to be a typo

    However the next line is wrong too

    x^2\le a means -\sqrt a\le x\le \sqrt a

    AS I already said, its a 2 to 1 transformation.
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  5. #5
    MHF Contributor matheagle's Avatar
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    I'll fix your error...

    F_{Z}(z)=P(Z\leq z)
    =P((X+1)^2\leq z)
    =P(-\sqrt{z}\leq X+1\leq\sqrt{z})

    now continue

    =P(-\sqrt{z}-1\leq X\leq\sqrt{z}-1)

    =F_X(\sqrt{z}-1)-F_X(-\sqrt{z}-1)

    NOW plug into the normal and differentiate
    Last edited by matheagle; November 28th 2009 at 09:02 AM.
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  6. #6
    Senior Member chella182's Avatar
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    I know, X^2+1 isn't the same as (X+1)^2 it was indeed a typo was trying to be quick 'cause the homework was due in on the day I replied. It was an unassessed question so it didn't matter really, was just intrigued.
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