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Beta/F Distribution

  • Thread starter cse63146
  • Start date
  • #1
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Homework Statement



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

Homework Equations





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:

Answers and Replies

  • #2
392
0
If W is F(m,n), then 1/W is F(n,m).
 
  • #3
452
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Sorry, but I'm still having problems with the tranformation.
 
  • #4
392
0
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]
 
  • #5
452
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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]
 
  • #6
392
0
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}[/tex]
Yes. Since 1/W is F(r2,r1), you are done.

[tex] = \frac{\frac{r_1}{r_2}W}{\frac{r_1}{r_2}W + 1} [/tex]
No.
 
  • #7
452
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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] ?
 
  • #8
392
0
I would substitute X for 1/W.
 
  • #9
452
0
Thanks for the help.
 

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