Beta/F Distribution: Show Y has Beta Dist.

[cse63146]

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

Homework Equations

The Attempt at a Solution

I know that

and

, then Y has a beta distribution.

Not sure what to do next.

 

[Billy Bob]

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

 

[cse63146]

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

 

[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}

 

[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}

 

[Billy Bob]

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.

 

[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} ?

 

[Billy Bob]

I would substitute X for 1/W.

 

[cse63146]

Thanks for the help.