Beta/F Distribution

1. Sep 23, 2009

cse63146

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

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

Not sure what to do next.

Last edited by a moderator: Apr 17, 2017
2. Sep 23, 2009

Billy Bob

If W is F(m,n), then 1/W is F(n,m).

3. Sep 23, 2009

cse63146

Sorry, but I'm still having problems with the tranformation.

4. Sep 23, 2009

Billy Bob

Just multiply the numerator and denominator of $$\frac{1}{1 + \frac{r_1}{r_2}W}$$ by an appropriate quantity, to put it in the form $$\frac{\frac{\nu_1}{\nu_2}X}{\frac{\nu_1}{\nu_2}X + 1}$$

5. Sep 23, 2009

cse63146

So:

$$\frac{1}{1 + \frac{r_1}{r_2}W}\frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1}\frac{1}{W}} = \frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1}\frac{1}{W} + 1} = \frac{\frac{r_1}{r_2}W}{\frac{r_1}{r_2}W + 1}$$

6. Sep 23, 2009

Billy Bob

Yes. Since 1/W is F(r2,r1), you are done.

No.

7. Sep 23, 2009

cse63146

so I can just leave it as $$\frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1} \frac{1}{W} + 1}$$ ?

8. Sep 24, 2009

Billy Bob

I would substitute X for 1/W.

9. Sep 24, 2009

cse63146

Thanks for the help.