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Beta/F Distribution

  1. Sep 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Let Y = [tex]\frac{1}{1 + \frac{r_1}{r_2}W}[/tex] and W ~ F(r1,r2). Show that Y has a beta distributoin

    2. Relevant equations



    3. The attempt at a solution

    I know that fa9e3f936a51e5d79665430fbed0d961.png and e828dff3279f65f494946ea0e3d00d75.png , then Y has a beta distribution.

    Not sure what to do next.
     
    Last edited by a moderator: Apr 17, 2017
  2. jcsd
  3. Sep 23, 2009 #2
    If W is F(m,n), then 1/W is F(n,m).
     
  4. Sep 23, 2009 #3
    Sorry, but I'm still having problems with the tranformation.
     
  5. Sep 23, 2009 #4
    Just multiply the numerator and denominator of [tex]\frac{1}{1 + \frac{r_1}{r_2}W}[/tex] by an appropriate quantity, to put it in the form [tex]\frac{\frac{\nu_1}{\nu_2}X}{\frac{\nu_1}{\nu_2}X + 1}[/tex]
     
  6. Sep 23, 2009 #5
    So:

    [tex]\frac{1}{1 + \frac{r_1}{r_2}W}\frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1}\frac{1}{W}} = \frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1}\frac{1}{W} + 1} = \frac{\frac{r_1}{r_2}W}{\frac{r_1}{r_2}W + 1} [/tex]
     
  7. Sep 23, 2009 #6
    Yes. Since 1/W is F(r2,r1), you are done.

    No.
     
  8. Sep 23, 2009 #7
    so I can just leave it as [tex]\frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1} \frac{1}{W} + 1}[/tex] ?
     
  9. Sep 24, 2009 #8
    I would substitute X for 1/W.
     
  10. Sep 24, 2009 #9
    Thanks for the help.
     
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