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Math Help - t distribution

  1. #1
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    t distribution

    It was given that a random variable T is said to have a tn distribution when:

    T=\frac{Z}{\sqrt{U/n}}
    but while proving for the density function (pdf) of a tn distribution, why is is equal to the joint distribution \int_{0}^{\infty}f_{T,U}(t,u).du?
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  2. #2
    MHF Contributor matheagle's Avatar
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    You first find the joint denisty of Z and U, where you needed to say that

    U\sim \chi^2_n and Z\sim N(0,1) (and they are indep.)

    Then pick a dummy varable, say W=Z or W=U.
    (IF you let W=U, then you can do this, but you better check your bounds of integration.)

    Use calc3 to find the density of T and W.

    THEN integrate out the W and you have the density of U.
    Last edited by matheagle; August 21st 2009 at 11:13 PM.
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  3. #3
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    Well initially I was trying to solve it by 1st letting:
    Y=\sqrt{U/n}
    f_Y(y)=f_u(ny^2)(2ny)

    then follow by:
    T=\frac{Z}{Y}
    f_T(t)=\int_{-\infty}^{\infty}|y|f_Y(y)f_z(yt).dy

    but it turned out to be very complicated
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  4. #4
    MHF Contributor matheagle's Avatar
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    this is in many books
    I bet you can find it online too.
    I did a quick search yesterday, but didn't find it.
    I have it some where in my office and I know I can do it,
    but I don't have the time.
    It's in Hogg and Craig.
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