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Math Help - joint / marginal pdf

  1. #1
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    Talking joint / marginal pdf

    Let the independent random variables X1 and X2 be N(0,1) and X^{2}(r), respectively. Ley Y1=\frac{X1}{\sqrt\frac{X^{2}}{r}} and Y2=X2.

    a) Find the joint pdf of Y1 and Y2
    b) Determine the marginal pdf of Y1 and show that Y1 has a t distribution

    Thanks!
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  2. #2
    Moo
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    Hello,

    I assume it's Y_1=\frac{X_1}{\sqrt{\frac{X_2^2}{r}}} ?

    Write down the joint pdf of (X_1,X_2), which is just the product of their pdf (because they're independent)

    Then, you can see that X_1=Y_1\cdot\sqrt{\frac{X_2^2}{r}}=Y_1\cdot\sqrt{\  frac{Y_2^2}{r}}=Y_1Y_2 \cdot r^{-1/2} (because Y_2=X_2>0)

    And then make the transformation
    T~:~ \mathbb{R}\times \mathbb{R}_+^* \to \mathbb{R}\times \mathbb{R}_+^*
    (x_1,x_2) \mapsto (y_1,y_2)=\left(\frac{x_1}{\sqrt{\frac{x_2^2}{r}}}  ,x_2\right)

    Also, don't forget to divide by the Jacobian !

    that's pretty "long" to do, so it'd be good for you and for us that you show what you've tried/done, what you want us to correct.
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  3. #3
    MHF Contributor matheagle's Avatar
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    This is okay. At first I thought you had two indep st. normals.

    You need this X_2 to be a \chi^2_r.

    I was lost on that missing subscript at first.
    The joint density of Y1 and Y2 should factor giving you independence.
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  4. #4
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    Red face

    I'm sorry. It is supposed to be Y_{1}=\frac{X_{1}}{\sqrt\frac{X_{2}}{r}} and I haven't a clue where to begin. Thanks
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  5. #5
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    Is this anywhere near what I am supposed to do?
    Thanks!
    Stats.pdf
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  6. #6
    MHF Contributor matheagle's Avatar
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    (3) is way off. you need to review jacobians from calculus 3.
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  7. #7
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    Thank you. I just have my statistics book though. Could you help point me in the right direction? I don't know what I did wrong based on my text.
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  8. #8
    MHF Contributor matheagle's Avatar
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    You should reread what Moo told you.
    Start with..Write down the joint pdf of , which is just the product of their pdf ..
    then obtain the jacobian and change variables.
    Here's a review that explains the change for polar, but it's the same idea.
    http://www.maths.abdn.ac.uk/~igc/tch...es/node77.html
    Here's some more calc 3...
    http://tutorial.math.lamar.edu/Class...Variables.aspx

    And this should be outlined in you stat book too.
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