# Confidence Intervals

• Apr 17th 2013, 12:43 PM
renolovexoxo
Confidence Intervals
Let Y1; Y2; : : : ; Yn be i.i.d. from a gamma distribution with known shape parameter alpha and unknown
scale parameter beta. Find a (1-alpha )% condfidence interval for the parameter . (Hint: the Minimum Variance Unbiased Estimator for is beta hat = Ybar/alpha ).

I'm having trouble setting this up. I know where to go once the statement
P(qgammalower<=Pivotal Quanitity<=qgammaupper). I can't seem to get the distribution right for this gamma distribution. Is it Ybar~gamma(shape=n, scale=beta/n)? I don't see how the estimator for beta hat comes in to play. Any help would be greatly appreciated.
• Apr 17th 2013, 08:43 PM
chiro
Re: Confidence Intervals
Hey renolovexoxo.

Hint: What is the distribution of the sum of independent Gamma random variables with those properties?
• Apr 18th 2013, 04:15 AM
renolovexoxo
Re: Confidence Intervals
is it gamma(shape=n, scale=Ybar/alpha)? Do i use the sum rather than the mean? I'm sorry, I'm not sure where to go exactly with the hint.
• Apr 18th 2013, 11:19 AM
renolovexoxo
Re: Confidence Intervals
I feel like I finished this wrong since I didn't use the MVUE, I've attached my work.
• Apr 18th 2013, 04:36 PM
chiro
Re: Confidence Intervals
According to Wikipedia site []Gamma distribution - Wikipedia, the free encyclopedia the distribution should be Gamma(n*beta,alpha/n*alpha) = Gamma(n*beta,1/n).

This will be your estimator if you what you gave in the original post is correct.