Suppose that a random variable Y has a gamma distribution w. parameters

and an unknown

. We have earlier proved that

has a chi-square distribution w. 4 df. Using

as a pivotal quantity. derive a 90% confidence interval for \beta.

I don't know if I'm on the right way. I create a new random variable

and look for

=

I then have a similar example in the book that tells me to set it up like this:

Must I derive the pdf of U manually or can I use that

is chi-square distributed and put in the pdf of a chi-square distribution?

That would be:

but with y=u and

= 4

I've tried to do that but I don't get the right answer, which is

.

Thanks for your help.