# Beta/F Distribution

## Homework Statement

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

## The Attempt at a Solution

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

Not sure what to do next.

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

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

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 $$\frac{\frac{\nu_1}{\nu_2}X}{\frac{\nu_1}{\nu_2}X + 1}$$

So:

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

So:

$$\frac{1}{1 + \frac{r_1}{r_2}W}\frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1}\frac{1}{W}} = \frac{\frac{r_2}{r_1}\frac{1}{W}}{\frac{r_2}{r_1}\frac{1}{W} + 1}$$
Yes. Since 1/W is F(r2,r1), you are done.

$$= \frac{\frac{r_1}{r_2}W}{\frac{r_1}{r_2}W + 1}$$
No.

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

I would substitute X for 1/W.

Thanks for the help.