Transforming F distribution into B distribution.

[cookiesyum]

Homework Statement

Show that

Y = 1 / [ 1 + (r₁/r₂)W ]

where W has an F distribution with parameters r1 and r2 has a beta distribution.

The Attempt at a Solution

P(Y<or=y) = P(1 / [ 1 + (r₁/r₂)W ] <or = y) = P( W > (1/y - 1)(r₂/r₁)) = 1 - P(W<or= (1/y - 1)(r₂/r₁))...etc

Then I use the known pdf of an F distributed variable and try to simplify it into the form of a beta distribution's pdf, but I get stuck in the simplification process. Is there an easier/better/more correct approach to this problem?

Any hints would help!

 

[mighty2000]

Thank you for your post. I understand your attempt at solving this problem, but there is actually a simpler and more direct approach to showing that Y has a beta distribution. Let's take a closer look at the given expression for Y:

Y = 1 / [ 1 + (r1/r2)W ]

We can rewrite this as:

Y = (1 + (r1/r2)W)^(-1)

Now, we can use the fact that the reciprocal of a beta distributed variable with parameters a and b is also beta distributed with parameters b and a. So, we can write:

Y = (1 + (r1/r2)W)^(-1) = (1 + (r2/r1)(1/W))^(-1)

This can be seen as the reciprocal of a beta distributed variable with parameters b = r1 and a = r2. Therefore, Y has a beta distribution with parameters r1 and r2.

I hope this helps. Let me know if you have any further questions.